THE 

PHILOSOPHY  OF  ARITHMETIC 

AS  DEVELOPED   FROM  THE 

THREE  FUNDAMENTAL  PKOCESSES 

OF 

SYNTHESIS,  ANALYSIS  AND  COMPARISON 

CONTAINING    ALSO 

A  HISTOET  OF  ARITHMETIC 
REVISED  EDITION. 


EDWARD  BROOKS,  Ph.  D., 

Superintendent  of  the  Public  Schools  of  Philadelphia. 

LATE  PRINCIPAL  OF  STATE  NORMAL  SCHOOL.  PENNSYLVANIA,  AND  AUTHOR  OF  , 
NORMAL  SERIES  OF  MATHEMATICS. 


The  highest  Science  is  the  greatest  simplicity." 


PHILADELPHIA : 

NORMAL    PUBLISHING    COMPANY. 
1904. 


Entered  according  to  Act  of  Congress,  in  the  year  1876,  by 

EDWARD  BROOKS, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


Copyright,  1901,  by  Edward  Brooks 


ELECTROTYPED  A  PRINTED 
BY 
THE  WICK  ERRH  AM  PRINTING  CO., 
LANCASTER,  PA. 


PREFACE. 


PROGRESS  in  education  is  one  of  the  most  striking  characteris- 
tics  of  this  remarkable  age.  Never  before  was  there  so  general 
an  interest  in  the  education  of  the  people.  The  development  of 
the  intellectual  resources  of  the  nation  has  become  an  object  of 
transcendent  interest.  Schools  of  all  kinds  and  grades  are  multi- 
plying in  every  section  of  the  country ;  improved  methods  of  train- 
ing have  been  adopted ;  dull  routine  has  given  way  to  a  healthy 
intellectual  activity ;  instruction  has  become  a  science  and  teach- 
ing a  profession. 

This  advance  is  reflected  in,  and,  to  a  certain  extent,  has  been 
pioneered  by,  the  improvements  in  the  methods  of  teaching  arith- 
metic. Fifty  years  ago,  arithmetic  was  taught  as  a  mere  collection 
of  rules  to  be  committed  to  memory  and  applied  mechanically  to 
the  solution  of  problems.  No  reasons  for  an  operation  were  given, 
none  were  required  ;  and  it  was  the  privilege  of  only  the  favored 
few  even  to  realize  that  there  is  any  thought  in  the  processes. 
Amidst  this  darkness  a  star  arose  in  the  East ;  that  star  was  the 
mental  arithmetic  of  Warren  Colburn.  It  caught  the  eyes  of  a  few 
of  the  wise  men  of  the  schools,  and  led  them  to  the  adoption  of 
methods  of  teaching  that  have  lifted  the  mind  from  the  slavery  of 
dull  routine  to  the  freedom  of  independent  thought.  Through  the 
influence  of  this  little  book,  arithmetic  was  transformed  from  a  dry 
collection  of  mechanical  processes  into  a  subject  full  of  life  and  in- 
terest. The  spirit  of  analysis,  suggested  and  developed  in  it,  runs 
to-day  like  a  golden  thread  through  the  whole  science,  giving  sim- 
plicity and  beauty  to  all  its  various  parts. 

(iii) 


238836 


IV  PREFACE. 

No  one  who  did  not  in  his  earlier  years  learn  arithmetic 
by  the  old  mechanical  methods,  and  who  has  not  experienced 
the  transition  to  the  new  analytic  ones,  can  realize  the  com- 
pleteness of  the  revolution  effected  by  this  little  work.  But  great 
as  has  been  its  influence,  it  should  be  remembered  that  it  does 
not  contain  all  that  is  essential  to  the  science  of  numbers.  Analysis 
in  its  mission,  has  done  all  that  it  was  possible  for  it  to  accomplish, 
but  it  is  not  sufficient  for  the  perfection  of  a  science.  There  must  be 
synthetic  thought  to  build  up,  as  well  as  analytic  thought  to  separate 
and  simplify.  Comparison  and  generalization  have  an  important 
work  to  perform  in  unfolding  the  relations  of  the  various  parts  and 
in  uniting  them  by  the  logical  ties  of  thought,  which  should  bind 
them  together  into  an  organic  unity.  What  we  now  need  for  the 
perfection  of  the  science  of  arithmetic  and  our  methods  of  teaching 
it,  is  a  more  philosophical  conception  of  its  nature,  and  a  logical 
relating  of  its  parts  which  analysis  leaves  in  a  disconnected  condition. 

It  is  worthy  of  remark  that  arithmetic,in  respect  to  logical  symme- 
try and  completeness,  differs  widely  from  its  sister  branch— geom- 
etry. The  science  of  geometry  came  from  the  Greek  mind  almost  as 
perfect  as  Minerva  from  the  head  of  Jove.  Beginning  with  definite 
ideas  and  self-evident  truths,  it  traces  its  way,  by  the  processes  of 
deduction,  to  the  profoundest  theorem.  For  clearness  of  thought, 
closeness  of  reasoning,  and  exactness  of  truths,  it  is  a  model  of  excel- 
lence and  beauty.  It  stands  as  a  type  of  all  that  is  best  in  the  classi- 
cal culture  of  the  thoughtful  mind  of  Greece.  Geometry  is  the  per- 
fection of  logic;  Euclid  is  as  classic  as  Homer. 

The  science  of  numbers,  originating  at  the  same  time,  seems  to 
have  presented  less  attractions  or  greater  difficulties  to  the  Greek 
mind.  It  is  true  that  the  great  thinkers  grew  enthusiastic  in  the 
contemplation  of  numbers,  and  spent  much  time  in  fanciful  specu- 
lations upon  their  properties,  but  this  did  comparatively  little  for 
the  development  of  the  science.  The  present  system  of  arithmetic 
is  mainly  the  product  of  the  thought  of  the  past  three  or  four  cen- 
turies.     Developed  by   minds  less  logical  than   those  of  the  old 


Greeks,  and  growing  partly  out  of  the  necessities  of  business,  it 
seems  not  to  have  acquired  that  scientific  exactness  and  finish 
which  belong  to  the  science  of  geometry.  That  it  has  intrinsically 
as  logical  a  basis  and  will  admit  of  as  logical  a  treatment,  cannot 
be  doubted.  To  endeavor  to  exhibit  the  true  nature  of  the  science, 
show  the  logical  relation  of  its  parts,  and  thus  aid  in  placing  it 
upon  a  logical  foundation  beside  its  sister  branch,  geometry,  is  the 
object  of  the  present  treatise. 

The  work  is  divided  into  five  parts,  besides  the  Introduction. 
The  Introduction  contains  a  Logical  Outline  of  Arithmetic,  and  a 
brief  History  of  the  science,  including  an  account  of  the  Origin  of 
the  Arabic  system,  the  Origin  of  the  Fundamental  Operations,  and 
an  account  of  the  Early  Writers  on  the  science.  The  facts  pre- 
sented have  been  gathered  from  a  variety  of  sources,  and  have 
been  carefully  compared,  so  far  as  was  possible,  with  the  originals, 
to  secure  entire  accuracy  in  the  statements.  The  principal  author- 
ities followed  are  Leslie,  Peacock,  De  Morgan,  Fink,  and  Ball. 
As  much  is  presented  as  it  is  supposed  will  be  of  interest  to  the 
teacher  or  general  reader  ;  any  who  desire  more  detailed  in- 
formation are  referred  to  the  writers  mentioned. 

Part  First  treats  of  the  general  nature  of  arithmetic,  embracing 
the  Nature  of  Number,  the  Nature  of  ArithmelUMl  Language,  and  the 
Nature  of  Arithmetical  Reasoning.  The  natute  of  Number  is  quite 
fully  considered,  especially  in  its  relation  to  the  idea  of  Time. 
Various  definitions  of  Number  are  presented  and  examined,  and 
the  effort  is  made  to  ascertain  that  which  may  be  regarded  as  the 
best  for  general  use. 

The  Nature  of  the  Language  of  Arithmetic  is  discussed  upon  a 
broader  basis  than  usual.  The  true  relation  of  Numeration  to 
Notation,  which  seems  to  have  been  overlooked  by  many  authors, 
and  which  is  frequently  not  understood  by  pupils,  is  explained. 
It  is  shown  that  Numeration  is  merely  the  oral  and  Notation  the  writ- 
ten language  of  Arithmetic.  The  philosophy  of  the  Arabic  system 
of  notation,  the  objections  to  the  decimal  scale,  and  the  advantages 
of  a  duodecimal  system  of  arithmetic,  are  discussed. 


Vi  PREFACE. 

Considerable  attention  is  given  to  the  nature  of  Arithmetical 
Reasoning,  a  subject  which  seems  not  to  have  been  very  clearly- 
understood  by  logicians  and  arithmeticians.  The  effort  is  made 
to  put  this  matter  upon  a  logical  basis,  and  to  ascertain  and  pre- 
sent the  true  nature  of  the  logical  processes  by  which  the  science 
of  numbers  is  unfolded.  The  ground  being  almost  entirely  new,  it 
is  not  to  be  supposed  that  the  investigation  is  at  all  complete ;  but 
it  is  hoped  that  what  is  given  may  induce  some  one  to  present  a 
more  thorough  development  of  the  subject. 

The  fundamental  idea  of  the  work  is  that  arithmetic  has  a  triune 
basis;  that  it  is  founded  upon  and  grows  out  of  the  three  logical 
processes,  Analysis,  Synthesis,  and  Comparison.  This  is  a  new  gen- 
eralization, and  is  believed  to  be  correct.  It  has  been  previously 
maintained  that  all  of  Arithmetic  is  contained  in  the  two  processes, 
Addition  and  Subtraction ;  and  that  the  whole  science  is  a  logical 
outgrowth  of  these  two  fundamental  ones.  In  this  work  it  is 
shown  that  Synthesis  and  Analysis  are  mechanical  operations,  giving 
rise  to  some  of  the  divisions  of  the  science,  that  the  mechanical 
processes  are  directed  by  the  thought  process  of  Comparison,  and 
that  this  itself  gives  rise  to  a  larger  part  of  the  science.  The  old 
writers  held  that  we  can  only  unite  and  separate  numbers ;  in  this 
work  it  is  held  that  we  can  unite,  separate,  and  compare  numbers. 

Proceeding  with  this  idea,  it  is  shown  that,  regarding  Addition, 
Subtraction,  Multiplication,  and  Division,  as  the  fundamental  oper- 
ations of  arithmetic,  there  will  arise  from  them  several  other  pro- 
cesses of  a  similar  character,  which  I  have  called  the  Derivative 
Processes  of  Synthesis  and  Analysis.  It  is  then  seen  that  for  each 
analytical  process  there  should  be  a  corresponding  synthetic  pro- 
cess. There  will  thus  arise  a  new  process,  the  opposite  of  Factoring, 
to  which  I  have  given  the  name  of  Composition.  This  process,  it 
will  be  seen,  contains  several  interesting  cases,  which  correlate  with 
the  different  cases  of  Factoring.  It  is  of  especial  interest  in  Alge- 
bra, as  may  be  seen  in  my  Elementary  Algebra. 

Continuing  this  thought,  it  is  shown  that  Ratio,  Proportion,  the 


PREFACE.  Vii 

Progressions,  etc.,  are  not  the  outgrowth  of  either  Synthesis  or 
Analysis,  but  of  the  thought  process — Comparison.  Attention  is 
called  to  the  nature  of  Ratio,  a  new  definition  is  suggested,  and  the 
correctness  of  the  prevailing  method  of  finding  the  ratio  of  two 
numbers,  which  has  been  questioned,  is  vindicated.  Suggestions 
are  also  made  for  improvements  in  some  of  the  definitions  and 
methods  of  treating  Ratio,  Proportion,  Progressions,  etc.  The  log- 
ical character  of  Percentage  is  exhibited,  and  the  simplest  and  most 
practical  method  of  treatment  suggested.  Several  interesting 
chapters  are  also  presented  upon  the  Theory  of  Numbers. 

The  subject  of  Fractions  is  quite  fully  discussed,  the  attempt  be- 
ing made  to  exhibit  their  nature  and  their  logical  relation  to  inte- 
gers. The  possible  cases  which  may  arise  are  considered,  and  a 
new  case,  called  the  Relation  of  Fractions,  first  given  in  one  of  my 
arithmetics,  and  already  introduced  into  several  other  arithmetical 
works,  is  presented  and  explained.  It  is  also  shown  that  the  sub- 
ject of  Fractions  admits  of  two  methods  of  treatment,  logically  distinct 
in  idea  and  form,  and  both  treatments  are  presented.  Especial  at- 
tention is  given  to  the  treatment  of  Circulates,  and  the  most  impor- 
tant principles  concerning  them  are  collated. 

The  nature  of  Denominate  Numbers,  which  seems  to  have  been 
imperfectly  understood,  is  explained  upon  what  is  regarded  as  the 
correct  basis.  They  are  shown  to  be  numerical  expressions  of  con- 
tinuous quantity,  in  which  some  artificial  unit  is  assumed  as  a  meas- 
ure. This  leads  to  the  adoption  of  a  new  definition  of  Denominate 
Numbers,  different  from  that  which  we  usually  find  in  our  text- 
books. The  origin  of  the  measures  in  the  various  classes  of 
Denominate  Numbers  is  also  stated,  and  many  interesting  facts 
concerning  them  are  given. 

While  the  philosophical  part  of  the  work  is  that  which  will  at- 
tract the  most  attention  among  thinkers,  the  historical  part  will  be 
quite  as  interesting  and  instructive  to  the  majority  of  younger 
readers.  In  the  historical  part,  of  course,  no  claims  to  original 
investigation  are  made ;  but  the  best  authorities  have  been  con- 


suited;  and,  in  many  cases,  their  very  language  has  been  used,  their 
expression  being  so  clear  and  concise  that  I  could  not  hope  to  im- 
prove it.  In  thus  combining  with  the  philosophy  of  arithmetic  its 
history,  which  in  many  cases  aids  in  unfolding  its  philosophy,  I 
have  aimed  to  present  a  work  especially  valuable  to  students  and 
the  younger  teachers  of  arithmetic.  Such  a  work,  I  feel,  would  have 
been  invaluable  to  me  in  my  earlier  years  as  a  teacher. 

It  is  proper  to  remark  that  the  work  was  mainly  written  about 
twelve  years  ago.  This  might  be  regarded  as  an  advantage;  for, 
according  to  the  recommendation  of  Horace,  publication  should  not 
be  hurried,  but  "  a  work  should  be  retained  till  the  ninth  year."  Quin- 
tilian  also  remarks  concerning  his  own  great  work  on  Oratory  that 
he  allowed  time  for  reconsidering  his  ideas,  "  in  order  that  when  the 
ardor  of  invention  had  cooled  I  might  judge  of  them  on  a  more 
careful  re-perusal,  as  a  mere  reader."  In  re-perusing  the  manuscript 
I  see  no  reason  for  any  change  of  opinion,  in  regard  to  any  of  the 
ideas  presented,  though  I  am  conscious  that  the  manner  of  pre- 
senting several  subjects  might,  in  some  respects,  be  improved  by 
being  re-written ;  but  I  have  decided  to  let  them  stand  as  originally 
conceived  and  expressed,  thinking  that  they  may  thus  gain  in  fresh- 
ness and  vividness  of  conception  what  they  may  lack  in  elegance  of 
style. 

Cherishing  many  pleasant  remembrances  associated  with  the 
discussion  of  these  ideas  before  my  pupils  in  the  class-room,  to 
many  of  whom  their  publication  will  prove  a  reminder  of  days 
gone  by,  I  commit  the  work,  with  its  merits  and  demerits,  to  an 
indulgent  public,  with  the  hope  that  it  may  be  of  assistance  to 
the  younger  members  of  the  profession,  and  contribute  somewhat 
towards  the  fuller  appreciation  of  the  interesting  and  beautiful 
science  of  numbers.  EDWARD  BROOKS. 

Normal  School,  3 filler svi lie,  Pa., 
January  16,  1876. 
I  revise  the  work   after  twenty-five   years,  giving  the   latest 
discoveries  in  the  history  of  arithmetic. 

EDWARD  BROOKS. 
Philadelphia,  May  20,  1901.  Supt.  Public  Schools. 


TABLE  OF  CONTENTS. 


INTRODUCTION. 


Chapter        I.  Logical  Outline  of  Arithmetic 9 

"  II.  Origin  and  Development  of  Arithmetic 17 

"  III.  Early  Writers  on  Arithmetic 29 

"  IV.  Origin  of  Arithmetical  Processes 44 

PART  I.— The  Nature  op  Arithmetic. 
Section  I. — The  Nature  of  Number. 

Chapter  I.  Number,  the  Subject-matter  of  Arithmetic 67 

"       II.  Definition  of  Number 72 

"     III.  Classes  of  Numbers 76 

"      IV.  Numerical  Ideas  of  the  Ancients 81 

Section  II. — Arithmetical  Language. 

Chapter       I.  Numeration,  or  the  Naming  of  Numbers 93 

"  II.  Notation,  or  the  Writing  of  Numbers 101 

"         III.  Origin  of  Arithmetical  Symbols 108 

"  IV.  The  Basis  of  the  Scale  of  Numeration, 113 

"  V.  Other  Scales  of  Numeration 121 

VI.  A  Duodecimal  Scale 126 

"        VII.  Greek  Arithmetic 135 

"      VIII.  Roman  Arithmetic 141 

"         IX.  Palpable  Arithmetic 147 

Section  III. — Arithmetical  Reasoning. 

Chapter      I.  There  is  Reasoning  in  Arithmetic 165 

"  II.  Nature  of  Arithmetical  Reasoning 171 

"         III.  Reasoning  in  the  Fundamental  Operations 177 

"  IV.  Arithmetical  Analysis 185 

"  V.  The  Equation  in  Arithmetic 193 

"  VI.  Induction  in  Arithmetic 197 

PART  II. — Synthesis  and  Analysis. 
Section  I. — Fundamental  Operations. 

Chapter      I.  Addition 207 

"  II.  Subtraction 213 

"         III.  Multiplication 221 

IV.  Division 227 

Section  II. — Derivative  Operations. 

Chapter      I.  Introduction  to  Derivative  Operations 237 

II.  Composition 240 

III.  Factoring 244 

IV.  Greatest  Common  Divisor 249 

V.  Least  Common  Multiple 257 

VI.  Involution 261 

VII.  Evolution 267 

1*  Ox) 


PAGK. 


X  CONTENTS. 

PART  III— Comparison. 
Section  I. — Ratio  and  Proportion. 

Chapter      I.  Introduction  to  Comparison 291 

"  II.  Nature  of  Ratio 294 

"         III.  Nature  of  Proportion 305 

IV.  Application  of  Simple  Proportion - .  .310 

"  V.  Compound  Proportion 318 

VI.  History  of  Proportion 326 

Section  II. — The  Progressions. 

Chapter      I.  Arithmetical  Progression 341 

"  II.  Geometrical  Progression 345 

Section  III. — Percentage. 

Chapter      I.  Nature  of  Percentage 355 

II.  Nature  of  Interest 361 

Section  IV. — The  Theory  of  Numbers. 

Chapter      I.  Nature  of  the  Subject 371 

"  II.  Even  and  Odd  Numbers  375 

"         III.  Prime  and  Composite  Numbers 378 

"  IV.  Perfect,  Imperfect,  etc.,  Numbers 383 

"  V.  Divisibility  of  Numbers 389 

"  VI.  The  Divisibility  by  Seven 397 

"        VII.  Properties  of  the  Number  Nine , 404 

PART  IV.— Fractions. 
Section  I. — Common  Fractions. 

Chapter      I.  Nature  of  Fractions 413 

"  II.  Classes  of  Common  Fractions 420 

"         III.  Treatment  of  Common  Fractions 426 

"  IV.  Continued  Fractions 434 

Section  II. — Decimal  Fractions. 

Chapter      I.  Origin  of  Decimals 443 

"  II.  Treatment  of  Decimals 455 

"         III.  Nature  of  Circulates 460 

"  IV.  Treatment  of  Circulates 464 

"  V.  Principles  of  Circulates 470 

"  VI.  Complementary  Repetends 476 

**        VII.  A  New  Circulate  Form 481 

PART  V.— Denominate  Numbers. 

Chapter      I.  Nature  of  Denominate  Numbers 489 

"  II.  Measures  of  Extension 497 

"  III.  Measures  of  Weight 512 

"  IV.  Measures  of  Value 521 

"  V.  Measures  of  Time 541 

•■  VI.  The  Metric  System 555 


INTRODUCTION 


TO    THE 


PHILOSOPHY  OF  ARITHMETIC 


I.  Logical  Outline  of  Arithmetic. 


II.  Origin  and  Development  of  Arithmetic. 


III.  Early  Writers  on  Arithmetic. 


IV.  Origin  of  Arithmetical  Processes. 


THE   PHILOSOPHY  OF  ARITHMETIC 


INTRODUCTION. 


CHAPTER  I. 

A   LOGICAL   OUTLINE   OF    ARITHMETIC. 

THE  Science  of  Arithmetic  is  one  of  the  purest  products  of 
human  thought.  Based  upon  an  idea  among  the  ear- 
liest which  spring  up  in  the  human  mind,  and  so  intimately 
associated  with  its  commonest  experience,  it  became  in- 
terwoven with  man's  simplest  thought  and  speech,  and  was 
gradually  unfolded  with  the  development  of  the  race.  The 
exactness  of  its  ideas,  and  the  simplicity  and  beauty  of  its  re- 
lations, attracted  the  attention  of  reflective  minds,  and  made 
it  a  familiar  topic  of  thought ;  and,  receiving  contributions  from 
age  to  age,  it  continued  to  develop  until  it  at  last  attained 
to  the  dignity  of  a  science,  eminent  for  the  refinement  of  its 
principles  and  the  certitude  of  its  deductions. 

The  science  was  aided  in  its  growth  by  the  rarest  minds  of 
antiquity,  and  enriched  by  the  thought  of  the  profoundest 
thinkers.  Over  it  Pythagoras  mused  with  the  deepest  enthu- 
siasm ;  to  it  Plato  gave  the  aid  of  his  refined  speculations ;  and 
in  unfolding  some  of  its  mystic  truths,  Aristotle  employed  his 
peerless  genius.  In  its  processes  and  principles  shines  the 
thought  of  ancient  and  modern  mind — the  subtle  mind  of  the 
Hindoo,  the  classic  mind  of  the  Greek,  the  practical  spirit  of 
the  Italian  and  English.     It  comes  down  to  us  adorned  with 

(9) 


10  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  offerings  of  a  thousand  intellects,  and  sparkling  with  the 
o-ems  of  thought  received  from  the  profoundest  minds  of  nearly 
every  age. 

And  yet,  rich  as  have  been  the  contributions  of  the  past, 
few  of  the  great  thinkers  have  endeavored  to  unfold  its  logical 
relations  as  a  science,  and  discover  and  trace  the  philosophic 
thread  of  thought  that  binds  together  its  parts  into  a  complete 
and  systematic  whole.  Unlike  its  sister  branch  geometry, 
which  came  from  the  Greek  mind  so  perfect  in  its  symmetry 
and  classic  in  its  logic,  the  science  of  arithmetic  has  been  treated 
too  much  as  a  system  of  fragments,  without  the  attempt  to 
coordinate  its  parts  and  weave  them  together  with  the  thread  of 
logic  into  a  complete  unity.  To  remedy  this  defect  is  the  special 
object  of  a  work  on  the  Philosophy  of  Arithmetic,  and  is  the 
task  which  the  author  of  the  present  work  has  with  diffidence 
attempted. 

Like  all  science,  which  is  an  organic  unity  of  truths  and 
principles,  the  science  of  arithmetic  has  its  fundamental  ideas, 
out  of  which  arise  subordinate  ones,  which  themselves  give 
rise  to  others  contained  in  them,  and  all  so  related  as  to  give 
symmetry  and  proportion  to  the  whole.  What  are  these  fun- 
damental and  derivative  ideas,  what  is  the  law  of  their  evolu- 
tion, what  is  the  philosophical  character  of  each  individual 
process,  and  what  is  the  logical  thread  of  thought  that  binds 
them  all  together  into  an  organic  unity  ?  These  are  the  ques- 
tions that  meet  us  at  the  threshold  of  the  effort  to  unfold  a 
philosophy  of  arithmetic ;  they  are  the  foundation  upon  which 
such  a  superstructure  must  be  erected  ;  and  we  begin  the 
answer  to  these  questions  in  the  first  chapter,  under  the  head 
of  A  Logical  Outline  of  Arithmetic,  which  exhibits  the  fun- 
damental operations  and  divisions  of  the  science. 

To  this  Logical  Outline  the  special  attention  of  the  reader 
is  invited,  as  it  is  not  only  the  foundation  upon  which  the  au- 
thor has  builded,  but  also  the  frame-work  of  the  system.     In 


A    LOGICAL    OUTLINE    OF    ARITHMETIC.  11 

it  the  science  is  assumed  to  be  based  upon  the  three  processes 

Synthesis,  Analysis,  and  Comparison;  general  processes  in 
which  each  individual  process  must  have  its  root,  and  from 
which  it  is  developed.  This  generalization  marks  a  new 
departure  in  the  method  of  regarding  the  science,  and  the  re- 
lation of  its  parts ;  and  shows  the  incorrectness  of  opinions 
around  which  has  gathered  the  dust  of  centuries.  Our  first 
inquiry  is,  what  is  A  Logical  Outline  of  Arithmetic  f 

All  numerical  ideas  begin  with  the  Unit.  It  is  the  origin, 
the  basis  of  arithmetic.  From  it,  as  a  fundamental  idea, 
originate  all  numbers  and  the  science  based  upon  them.  Begin- 
ning, then,  at  the  Unit,  let  us  see  how  the  science  of  arithmetic 
originates  and  is  developed. 

The  Unit  can  be  multiplied  or  divided.  This  gives  rise  to 
two  classes  of  numbers,  Integers  and  Fractions.  Integers 
originate  in  a  process  of  synthesis,  Fractions  in  a  process  of 
analysis.  Each  Integer  is  a  synthetic  product  derived  from  a 
combination  of  units;  each  Fraction  is  an  analytic  product 
derived  from  the  division  of  the  unit.  There  are,  therefore, 
two  general  classes  of  numbers,  Integers  and  Fractions, 
treated  of  in  the  science  of  arithmetic. 

Having  obtained  numbers  by  a  combination  of  units,  we  may 
unite  two  or  more  numbers  and  thus  obtain  a  larger  number 
by  means  of  synthesis ;  or  we  may  reverse  the  operation  and 
descend  to  a  smaller  number  by  means  of  analysis.  Numbers, 
therefore,  can  be  united  together  and  taken  apart ;  they  can  be 
synthetized  and  analyzed;  hence  Synthesis  and  Analysis  are 
the  two  fundamental  operations  of  arithmetic.  These  funda- 
mental operations  give  rise  to  others  which  are  modifications 
or  variations  of  them.  Arithmetic,  therefore,  from  its  primary 
conception  seems  to  consist  of  but  two  things, — to  increase  and 
to  diminish  numbers,  to  unite  and  to  separate  them.  Its  pri- 
mary operations  are  Synthesis  and  Analysis. 

To  determine  when  and  how  to  unite,  and  when  and  how 


12  THE   PHILOSOPHY    OF    ARITHMETIC. 

to  separate,  we  employ  a  process  of  reasoning  called  Compari- 
son. This  process  compares  numbers  and  determines  their 
relations.  Synthesis  and  Analysis  are  mechanical  processes , 
Comparison  is  the  thought  process.  Comparison  directs  the 
original  processes,  modifies  them  so  as  to  produce  from  them 
new  ones,  and  also  gives  rise  to  other  processes  not  contained 
in  the  original  ones.  It  is,  in  other  words,  by  this  thought 
process  working  upon  the  idea  of  number,  that  the  original 
processes  of  Synthesis  and  Analysis  are  directed  and  modified, 
that  other  processes  are  developed  from  them,  and  that  new 
and  independent  processes  arise,  and  the  science  of  arithmetic 
is  developed.  Comparison,  therefore,  in  arithmetic  as  in  geom- 
etry, is  the  process  by  which  the  science  is  constructed,  or 
the  key  with  which  the  learner  unlocks  its  rich  storehouse  of 
interest  and  beauty. 

Arithmetic,  it  is  thus  seen,  consists  fundamentally  of  three 
things;  Synthesis,  Analysis  and  Comparison.  Synthesis  and 
Analysis  are  fundamental  mechanical  operations,  suggested  in 
the  formation  of  numbers;  Comjiarison  is  the  fundamental 
thought  process  which  controls  these  operations,  unfolds 
their  potential  ideas,  and  also  gives  rise  to  other  divisions  of 
the  science  growing  immediately  out  of  itself.  In  other 
words,  the  science  of  arithmetic  has  a  triune  basis;  it  has  its 
roots  in,  and  grows  out  of,  the  three  processes,  Synthesis 
Analysis,  and  Comparison.  Let  us  examine  these  processes 
and  see  the  number,  nature,  and  relations  of  the  divisions 
growing  out  of  the  fundamental  operations,  and  thus  deter- 
mine the  logical  character  of  the  science  of  arithmetic. 

Synthesis. — A  general  synthesis  is  called  Addition.  A  spe- 
cial case  of  the  synthetic  process  of  Addition,  in  which  the 
numbers  added  are  all  equal,  their  sum  receiving  the  name  of 
product,  is  called  Multiplication.  The  forming  of  Composite 
Numbers  by  a  synthesis  of  factors,  which  may  be  called 
Composition ;  Multiples,  formed  by  a  synthesis  of  particular 
factor? ;  and  Involution,  by  a  synthesis  of  equal  factors,  are 


A  LOGICAL  OUTLINE    OF    ARITHMETIC.  13 

all  included  under  Multiplication.  Hence,  since  Involution, 
Multiples,  and  Composition,  are  special  cases  of  Multiplication, 
and  Multiplication  is  itself  a  special  case  of  Addition,  the  pro- 
cess of  Addition  includes  all  the  synthetic  processes  to  which 
numbers  can  be  subjected. 

Analysis. — A  general  analysis,  the  reverse  of  Addition,  is 
called  Subtraction.  A  special  case  of  Subtraction,  in  which 
the  same  number  or  equal  numbers  are  successively  subtracted 
with  the  object  of  ascertaining  how  many  times  the  number 
subtracted  is  contained  in  another,  is  called  Division.  Factor- 
ing is  a  special  case  of  Division  in  which  many  or  all  of  the 
factors  of  a  number  are  required ;  Evolution  is  a  special  case 
of  factoring  in  which  one  of  the  several  equal  factors  is  re- 
quired ;  and  Common  Divisor  is  a  case  of  factoring  in  which 
some  common  factor  of  several  numbers  is  required.  The 
process  of  Division,  therefore,  includes  the  processes  of  Factor- 
ing, Common  Divisor,  and  Evolution;  and  since  Division  is  a 
special  case  of  Subtraction,  all  of  these  processes  are  logically 
included  under  the  general  analytic  process  of  Subtraction. 

Comparison. — By  comparison  the  general  notion  of  relation 
is  attained,  out  of  which  arise  several  distinct  arithmetical 
processes.  By  comparing  numbers,  we  perceive  the  relations 
of  difference  and  quotient;  and  giving  measures  to  these,  we 
have  Ratio.  A  comparison  of  equal  ratios  gives  us  Propor- 
tion. A  comparison  of  several  numbers  differing  by  a  common 
ratio  gives  us  Arithmetical  and  Geometrical  Progression.  In 
comparing  concrete  numbers,  when  the  unit  is  artificial,  we 
perceive  that  they  differ  in  regard  to  the  value  of  the  units, 
and  also  that  we  can  change- a  number  of  units  of  one  species 
into  a  number  of  another  species  of  the  same  class  ;  and  thus 
we  have  the  process  called  Reduction.  In  comparing  abstract 
numbers  we  notice  certain  relations  and  peculiarities  which, 
investigated,  give  rise  to  the  Properties  or  principles  of  num- 
bers. In  comparing  numbers,  we  may  assume  some  number 
as  a  basis  of  reference  and  develop  their  relations  in  regard  to 


14  THE   PHILOSOPHY    OF    ARITHMETIC. 

this  basis ; — when  this  basis  is  a  hundred,  we  have  the  pro- 
cess called  Percentage. 

Thus  we  obtain  a  complete  outline  of  the  science  of  numbers, 
and  perceive  more  clearly  the  logical  relations  of  the  divisions 
of  the  science.  Arithmetic  is  conceived  as  based  upon  the  two 
fundamental  operations,  synthesis  and  analysis,  these  opera- 
tions being  controlled  by  comparison,  which  develops  new 
processes  from  these  and  also  from  itself.  The  whole  science 
of  Pure  Arithmetic  is  the  outgrowth  of  this  triune  basis,  Syn- 
thesis, Analysis,  and  Comparison.  The  rest  of  arithmetic 
consists  of  the  solution  of  problems,  either  real  or  theoretic, 
and  may  be  included  under  the  head  of  Applications  of  Arith- 
metic. 

This  conception  of  the  subject  is  new  and  important.  It  has 
been  heretofore  held  that  addition  and  subtraction  compre- 
hended the  entire  science  of  arithmetic;  that  all  other  pro- 
cesses are  contained  in  them,  and  are  an  outgrowth  from  them. 
This  is  a  fallacy,  which,  among  other  things,  has  led  logicians 
to  the  absurd  conclusion  that  there  is  no  reasoning  in  arith- 
metic. Assuming  that  there  is  no  reasoning  in  the  primary 
processes  of  synthesis  and  analysis,  and  that  these  primary 
processes  contain  the  entire  science,  they  naturally  conclude 
that  there  is  no  reasoning  in  the  science  itself.  The  analysis 
of  the  subject  here  given  dispels  this  error  and  exhibits  the 
subject  in  its  true  light.  Synthesis  and  Analysis  are  seen  to 
be  the  primary  mechanical  processes;  Comparison,  the  thought 
process,  touches  them  with  her  wand  of  magic,  and  they  ger- 
minate and  bring  forth  other  processes,  having  their  roots  in 
these  primary  ones.  Comparison  also  becomes  the  foundation 
of  processes  distinct  from  those  of  synthesis  and  analysis, 
processes  which  cannot  be  conceived  as  growing  out  of  syn- 
thesis and  analysis,  but  which  have  their  root  in  the  thought 
process  of  the  science — in   Comparison. 

This  outline  of  the  science  grows  out  of  the  pure  idea  of 
number,  independent  of  the    language   of  arithmetic.     These 


A   LOGICAL    OUTLINE   OF    ARITHMETIC. 


15 


fundamental  processes  are  modified  by  the  method  of  notation 
employed  to  express  numbers.  With  the  Roman  or  Greek 
methods  of  notation,  the  methods  of  operation  would  not  be 
the  same  as  with  the  Arabic  system.  The  method  of  "  carry- 
ing one  for  every  ten,"  of  "borrowing"  in  subtracting,  the 
peculiar  methods  of  multiplying  and  dividing,  grow  out  of  the 
Arabic  system  of  notation.  A  portion  of  the  treatment  of 
common  and  decimal  fractions  arises  from  the  notation  adopted, 
and  the  principles  and  processes  of  repetends  originate  in  the 
same  manner.  The  methods  of  extracting  square  and  cube 
root  would  be  different  if  we  employed  a  different  method  of 
expressing  numbers.  It  is  thus  seen  that  the  fundamental 
divisions  of  arithmetic  arise  from  the  pure  idea  of  number,  that 
the  processes  in  these  divisions  are  modified  by  the  method  of 
notation  adopted,  and  also  that  some  of  the  principles  and  pro- 
cesses of  the  science  grow  out  of  this  notation.  It  may  be 
remarked,  also,  that  the  power  of  arithmetic  as  a  calculus 
depends  upon  the  beautiful  and  ingenious  system  of  notation 
adopted  to  express  numbers. 

It  is  believed  that  the  above  view  of  arithmetic  must  tend 
to  simplify  the  subject,  and  that  much  clearer  notions  of  the 
science  will  be  attained  when  these  philosophical  relations  are 
apprehended.  A  general  view  of  the  subject  is  presented  by 
the  following  analytical  outline  : 

t    CTTT,fV,^o,-a  /Addition.  ( Composition. 

±.  byntnesis.  ^  Multiplication>  J     7  Common  Multiple. 

(    \  Involution. 


Logical 

Outline 

of 

Arithmetic. 


„  f  Subtraction 
II.  Analysis.  {Division. 


III.  Comparison. 


{Factoring, 
f  Common  Divisor. 
\  Evolution. 

1.  Ratio. 

2.  Proportion. 

3.  Progression. 

4.  Reduction. 

5.  Percentage. 

6.  Propertied  of  Numbers. 


16 


THE   PHILOSOPHY   OF   AKITHMETIC. 


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CHAPTER  II. 

ORIGIN    AND    DEVELOPMENT    OF    ARITHMETIC. 

A  knowledge  of  Arithmetic  is  coeval  with  the  race.  Every 
■*"*-  people,  no  matter  how  uncivilized,  must  have  possessed  some 
ideas  of  numbers,  and  employed  them  in  their  transactions  with 
one  another.  These  ideas  would  be  multiplied,  and  the  methods 
of  operation  founded  upon  them  gradually  extended  and  improved 
as  the  nation  advanced  in  civilization  and  intelligence.  The  his- 
tory of  Arithmetic  is,  therefore,  inseparably  connected  with  the 
history  of  civilization  and  the  race.  The  origin  of  its  elementary 
processes  must,  of  necessity,  be  involved  in  obscurity  and  uncer- 
tainty. History  can  speak  positfvely  only  of  some  of  the  higher 
and  more  recent  developments  of  the  science. 

In  presenting  some  of  the  principal  facts  concerning  the  history 
of  arithmetic,  we  shall  consider  three  things :  the  origin  of  our 
present  system  of  arithmetic  ;  the  early  writers  on  the  science ;  and 
the  origin  of  the  fundamental  operations.  %  Other  historical  facts 
will  be  mentioned  in  connection  with  the  particular  subjects  to 
which  they  belong.  One  of  the  most  interesting  inquiries  is  that 
which  relates  to  the  origin  of  the  system  of  arithmetic  now  gen- 
erally adopted,  which  we  shall  consider  in  the  present  chapter. 

The  basis  of  our  system  of  arithmetic  is  the  principle  of  place- 
value  in  writing  numbers.  All  civilized  nations,  from  the  primi-  \- 
tive  habit  of  reckoning  with  the  fingers,  adopted  a  system  of  count- 
ing by  groups  of  ten.  Each  group  of  ten  is  distinguished  by  a 
special  name,  and  the  names  of  the  first  nine  numbers  are  used  to 
^number  the  groups  and  express  the  numbers  between  them.  Thus 
all  civilized  peoples  adopted  the  same  general  method  of  oral 
arithmetical  language.  In  writing  numbers,  however,  different 
2  (17) 


18  THE   PHILOSOPHY   OF   ARITHMETIC. 

nations  adopted  widely  different  methods  of  notation.  Our  present 
simple  and  practical  system  of  notation  was  reached  by  only  a 
single  nation  of  antiquity.  The  various  methods  of  writing  num- 
bers in  use  among  the  ancient  nations  and  the  origin  of  our  present 
system  will  be  briefly  considered. 

The  Egyptians  represented  numbers  by  written  words,  and  also 
by  symbols  for  each  unit  repeated  as  often  as  necessary.  In  one 
of  the  tombs  near  the  pyramid  of  Gizeh,  hieroglyphic  numerals 
have  been  found  in  which  1  is  represented  by  a  vertical  line;  10 
by  a  kind  of  horse-shoe  ;  100  by  a  short  spiral ;  10,000  by  a  point- 
ing finger  ;  100,000  by  a  frog,  and  1,000,000  by  the  figure  of  a 
man  in  the  attitude  of  wonder.  In  their  hieratic  writing  they  used 
symbols  for  numbers,  but  they  did  not  combine  them  on  the  prin- 
ciple of  place-value  as  in  the  modern  system  of  notation.  There 
are  special  characters  for  the  nine  units,  and  also  for  the  tens,  the 
hundreds,  etc.     The  following  are  specimens  of  these  symbols  : 

I  U  III  -  "1  A  A  (A  -^  "1  iU  1 

1         2  3  4  5  10  20  30  40  50        60  70 

These  are  combined  on  the  additive  principle,  the  symbol  of  the 
larger  value  always  being  placed  at  the  left  of  that  for  the  smaller 
value.  The  papyrus  of  Ahmes  in  the  British  Museum  indicates 
that  the  Egyptians  at  a  very  early  period  had  considerable  knowl- 
edge of  the  art  of  arithmetic. 

The  ancient  Babylonians  used  the  wedge-shaped  characters  of 
their  cuneiform  system  of  writing  in  the  representation  of  numbers. 
The  mark  for  unity,  a  vertical  arrow  head,  is  repeated  up  to  ten, 
whose  symbol  is  a  barbed  sign  pointing  to  the  left.  These  symbols 
by  mere  repetition  served  to  express  numbers  up  to  one  hundred, 
for  which  a  new  sign  was  employed.  The  characters  were  written 
sometimes  one  beside  another,  and  sometimes,  to  save  space,  one 
over  another.  The  symbol  for  the  smaller  number  written  to  the 
right  of  the  symbol  for  a  hundred  denoted  addition  ;  the  same  sym- 
bol written  on  the  left  denoted  multiplication,  or  the  number  of 
hundreds.  The  Babylonians  thus  employed  the  principle  of  place 
value  to  some  extent,  but  having  no  symbol  for  zero  they  were  una- 


ORIGIN   OF   THE   SCIENCE.  19 

ble  to  develop  the  modern  system  of  notation  and  calculation.  They 
used  also  along  with  the  decimal  system  the  sexagesimal  system,  that 
is  one  with  a  base  of  60,  and  their  operations  with  both  integers 
and  fractions  show  considerable  mathematical  facility  and  skill. 

The  Chinese  had  a  well  developed  number  system,  and  seem  to 
have  come  as  near  the  present  method  of  notation  as  any  nation  of 
antiquity,  except  the  Hindoos.  Of  their  early  number  symbols  but 
little  seems  to  be  known.  Later,  as  a  result  of  foreign  influence, 
there  arose  two  new  kinds  of  notation,  whose  figures  are  supposed 
to  resemble  the  ancient  symbols.  Though  the  Chinese  wrote  their 
word-symbols  in  columns,  yet  their  numbers  were  written  from 
left  to  right,  beginning  with  the  highest  order.  The  ordinal  and 
cardinal  numbers  are  usually  arranged  in  two  lines,  one  above 
another,  with  zeros  in  the  form  of  small  circles  appearing  as  often 
as  necessary.     The  following  symbols  will  illustrate  this  system : 

II      X      4.      +      fi  o 

2  4  6  10  10,000  0 

"  X 

fiOO  -*-_,_ 

20,046 

Their  arithmetical  calculations  were  made  by  means  of  the 
■abacus  or  swan-pan,  which  is  used  at  the  present  day  among  both 
their  scholars  and  their  merchants. 

The  Phoenicians  expressed  numbers  in  words,  and  also  by  the 
use  of  special  numerical  symbols,  using  vertical  marks  for  the 
units  and  horizontal  marks  for  the  tens.  The  Syrians  somewhat 
later  used  the  twenty-two  letters  of  their  alphabet  to  represent  the 
numbers  1,2,.  .  .  9,  10,  20,  .  .  .  90,  100,  .  .  .  400  ;  500  was 
400+  100,  etc.  The  thousands  were  represented  by  the  symbols 
for  units  with  a  subscript  comma  at  the  right.  The  notation  of 
the  Hebrews  followed  the  same  plan.  None  of  these  nations  had 
a  notation  that  could  be  used  in  making  calculations  as  we  do 
with  the  modern  system. 

The  early  Greeks  seem  to  have  used  the  initial  letters  of  their 


20  THE   PHILOSOPHY    OF   ARITHMETIC. 

number  words  to  represent  written  numbers ;  as  [""]  for  5  (iret>re)f 
A  for  10  (<5f/ca),  and  these  letters  were  repeated  as  often  as  neces- 
sary. Soon  after  500  B.  C.  two  new  systems  appeared  among  the 
Greeks.  One  used  the  24  letters  of  the  Ionic  alphabet  in  their 
natural  order  for  the  numbers  from  1  to  24.  The  other  arranged 
these  letters,  together  with  three  other  symbols,  in  an  arbitrary 
order,  thus  «=1,  P  =  2,  .  .  .  i=  10,  «==  20,  .  .  .  p=100,  *= 
200,  etc.  The  Greeks  could  perform  the  fundamental  operations 
with  these  symbols  with  considerable  facility,  as  may  be  seen  in 
the  chapter  on  Greek  arithmetic.  The  common  method  of  calcu- 
lation, however,  seems  to  have  been  with  the  abacus.  The  Greeks 
did  not  make  use  of  the  principle  of  place  value,  and  they  had  no 
symbol  for  zero. 

The  Romans  also  expressed  numbers  by  means  of  letters. 
The  characters  are  supposed  to  have  been  inherited  from  the 
Etruscans,  and  may  originally  have  been  symbolic,  and  subse- 
quently, on  account  of  their  resemblance  to  forms  in  their  alpha- 
bet, they  were  replaced  by  letters.  Mommsen  says  that  the  Roman 
numerals  I,  V,  X  represent  the  finger,  the  hand,  and  the  double 
hand  respectively.  These  characters  were  combined  according  to 
the  additive  principle,  as  in  VI,  VII,  VIII,  and  also  in  accord- 
ance with  the  subtractive  principle,  as  in  IV,  IX,  XL,  XC.  This 
subtractive  principle  is  a  distinctive  characteristic  of  the  Roman 
system  of  notation.  The  Romans  could  not  use  their  notation  for 
reckoning,  but  made  their  calculations  with  counters  (calculi)  or 
with  the  abacus.  They  seem  to  have  had  no  conception  of  place 
value,  or  of  a  symbol  for  zero,  as  their  system  did  not  call  for  it. 

Our  present  number  system,  it  is  now  known,  had  its  origin 
among  the  Hindoos.  They  originated  the  modern  position-system, 
and  introduced  the  zero  to  fill  an  unoccupied  place.  Their 
so-called  sacred  books  which  have  been  in  the  hands  of  the  priest- 
hood for  centuries,  contain  the  numerical  characters.  Their  ear- 
liest symbols  of  the  nine  digits  were  after  3  merely  abridged  num- 
ber words,  and  the  use  of  letters  as  figures  is  said  to  date  from  the 
second  century  B.  C.     The  development  of  the  system  of  place- 


ORIGIN   OF   THE   SCIENCE.  21 

value  seems  to  have  proceeded  slowly.  In  an  early  method  of 
writing  numbers  there  is  no  indication  of  the  principle  of  place- 
value,  though  it  appears  in  two  other  systems  of  notation  which 
prevailed  in  Southern  India.  Both  of  these  methods  are  dis- 
tinguished by  the  fact  that  the  same  number  can  be  made  up  in 
various  ways.  One  method  consisted  in  employing  the  alphabet, 
in  groups  of  nine  symbols,  to  denote  the  numbers  from  1  to  9  re- 
peatedly, while  certain  vowels  denote  the  zeros.  A  second 
method  used  type-words,  and  combined  them  according  to  the  law 
of  position.  Thus  ubdhi  (one  of  the  4  seas)  =  4  ;  surya  (the  sun 
with  its  12  houses)  =  12  ;  agvin  (the  two  sons  of  the  sun)  =  2. 
The  combination  abdhisuryagvinas  denoted  the  number  2124. 
These  no  doubt  were  stepping-stones  to  the  present  simple  appli- 
cation of  the  position  principle.  The  modern  system  of  place 
value  could  not  have  been  adopted  before  the  invention  of  the 
zero,  and  there  is  no  proof  of  its  being  introduced  before  400  A.  D. 
The  first  known  use  of  the  symbol  on  a  document,  Cantor  says, 
dates  from  738  A.  D. 

The  Arabs  became  acquainted  with  the  Hindoo  number-system 
and  its  figures,  including  zero,  in  the  eighth  century,  and  were 
instrumental  in  introducing  the  system  into  Europe.  It  was  for 
many  years  thought  that  the  present  system  of  arithmetic  origi- 
nated with  the  Arabians.  The  characters  in  general  use  were 
called  Arabic  characters,  and  the  method  of  writing  numbers  was 
known  as  the  Arabic  system  of  notation.  Further  proof,  it  was 
thought,  was  found  in  the  two  words  "cipher"  and  "  zero," 
cipher  being  the  Arabic  as-sifr,  meaning  empty.  This  word, 
however,  was  derived  no  doubt  from  the  Sanskrit  name  of  the 
naught,  sunya,  the  void.  In  Italy  the  character  for  naught  was 
called  "  zephiro,"  which  has,  by  rapid  pronunciation,  been  changed 
to  zero.  The  Arabs,  however,  it  is  now  known,  were  not  the 
authors  of  the  system,  but  derived  it  from  the  Hindoos,  and  were 
only  instrumental  in  introducing  it  into  Europe. 

The  Arabs  from  an  early  period  had  commercial  relations  with 
India,  which  brought  them  in  contact  with  the  Indian  system  of 


22  THE   PHILOSOPHY   OF   ARITHMETIC. 

reckoning.  It  is  known  that  they  were  acquainted  with  the 
Hindoo  number  system  and  its  figures,  including  the  zero,  as 
early  as  the  eighth  century.  The  earliest  definite  date,  says  Ball, 
assigned  for  the  use  in  Arabia  of  the  decimal  system  of  notation 
is  773.  In  that  year  some  Indian  astronomical  tables  were 
brought  to  Bagdad  in  which  it  is  almost  certain  the  Indian 
numerals,  including  the  zero,  were  used.  The  Arabs  no  doubt 
developed  the  system  somewhat  slowly,  as  the  custom  of  writing 
out  number  words  continued  among  them  until  the  beginning  of 
the  eleventh  century.  In  the  investigation  we  meet  with  the 
singular  fact  that  the  Arabs  employed  two  kinds  of  figures  :  one 
used  chiefly  in  the  East  called  "  Oriental ;"  another  used  by  the 
Western  Arabs  in  Africa  and  Spain  called  the  Gubar  or  dust 
numerals,  so  called  because  they  were  first  introduced  among  the 
Arabs  by  an  Indian  who  used  a  table  covered  with  fine  dust  for  the 
purpose  of  ciphering.  These  Gubar  numerals  are  the  ancestors 
of  our  modern  numerals.  They  are  said  to  be  modifications  of  the 
initials  of  the  Sanskrit  word-numerals. 

It  was  through  Spain,  however,  it  is  generally  believed,  rather 
than  directly  from  Arabia,  that  the  Arabic  system  was  introduced 
into  Europe.  The  Moors  as  early  as  747  had  conquered  Spain, 
and  established  there  their  rule.  They  brought  with  them  a  taste 
for  learning,  and  established  schools  and  universities,  so  that  by 
the  tenth  and  eleventh  centuries  they  had  attained  to  a  high  degree 
of  civilization.  Though  the  political  relations  of  the  Arabs  with 
the  caliphs  of  Bagdad  were  not  entirely  cordial,  yet  they  gave 
ready  welcome  to  the  works  of  the  great  Arabian  mathematicians. 
The  Arabs  had  studied  with  great  avidity  the  Greek  mathematics, 
and  their  translations  of  Euclid,  Archimedes,  Ptolemy,  etc.,  along 
with  the  works  of  the  Arabians  themselves  on  arithmetic  and 
algebra,  were  studied  at  the  great  Moorish  universities  of  Gren- 
ada, Cordova  and  Seville. 

Thus  while  the  Christian  world  was  enveloped  in  ignorance, 
the  Arabs  were  cultivating  the  learning  and  literature  of  Greece. 
Though    not    highly    gifted   with   creative   powers   of  mind  by 


ORIGIN   OF   THE   SCIENCE.  23 

which  they  made  many  valuable  additions  to  what  they  thus 
acquired,  a  debt  of  gratitude  is  due  them  because  they  "  preserved 
and  fanned  the  holy  fire."  Their  efforts  at  conquest  had  been 
crowned  with  brilliant  success.  Spain  had  yielded  to  their  sway, 
and  the  Moors  had  become  celebrated  throughout  Europe  for  the 
splendor  of  their  institutions,  the  magnificence  of  their  architecture, 
and  the  proficiency  of  their  scholars. 

Disgusted  with  the  trifling  of  their  own  schools,  energetic  and 
aspiring  young  men  from  England  and  France  repaired  to  Spain 
to  learn  philosophy  from  the  accomplished  Moors.  There  they 
studied  arithmetic,  geometry  and  astronomy,  and  made  themselves 
familiar  with  the  Arabic  method  of  notation  and  calculation.  On 
their  return  they  brought  the  characters  and  methods  of  the 
Arabic  arithmetic  with  them  and  introduced  them  to  the  scholars 
of  Northern  Europe,  and  thus  in  time  they  gradually  displaced  the 
Roman  system,  which  had  been  in  use  for  many  centuries. 

One  of  these  "  pilgrims  of  science  "  was  an  obscure  monk  of 
Auvergne  named  Gerbert,  who  died  in  1003.  Returning  to  his 
native  country  he  became  widely  celebrated  for  his  genius  and 
learning,  and  subsequently  rose  to  the  papal  chair  with  the  title 
of  Svlvester  II.  His  treatises  on  arithmetic  and  geometry  were 
valuable,  presenting  many  rules  for  abbreviating  the  operations  in 
common  use.  He  introduced  an  improvement  in  the  use  of  the 
abacus  by  marking  each  of  the  nine  beads  in  every  column  with  a 
distinctive  sign.  These  marks,  called  apices,  are  supposed  to 
have  been  the  same  as  the  Gubar  numerals,  and  thus  Gerbert  did 
much  to  introduce  the  old  Hindoo  numeral- forms  into  Western 
Europe. 

Efforts  have  been  made  to  ascertain  what  persons  were  most 
conspicuous  in  the  introduction  of  the  Arabic  characters  into 
Northern  Europe.  There  seems  to  have  been  some  difficulty  in 
obtaining  access  to  the  Moorish  universities,  as  the  Moors  are  said 
to  have  taken  pains  to  conceal  their  learning  from  the  Christian 
world.  One  of  the  earliest  students  from  Christian  Europe  to 
acquire  a  knowledge  of  Moorish  aud  Arabian  science  was  an  Eng- 


24  THE   PHILOSOPHY   OF   ARITHMETIC. 

lish  monk,  Adelhard  of  Bath,  who,  disguised  as  a  Mahommedan 
student,  got  into  Cordova  about  1120  and  obtained  a  copy  of 
Euclid's  Elements.  This  copy  translated  into  Latin  is  said  to 
have  been  the  foundation  of  all  the  editions  of  the  work  known  in 
Europe  until  1533. 

Another  scholar  who  was  influential  in  the  introduction  of 
Moorish  learning  into  Northern  Europe  was  Abraham  Ben  Ezra, 
a  Jewish  rabbi,  born  at  Toledo  in  1097  and  died  at  Rome  in  1167. 
He  wrote  an  arithmetic  in  which  he  explains  the  Arabic  system 
of  notation  with  nine  symbols  and  a  zero,  and  gives  the  funda- 
mental processes  of  arithmetic  and  the  rule  of  three. 

Another  eminent  scholar  who  aided  in  the  introduction  was 
Gerard,  born  in  1114  and  died  in  1187.  He  translated  the  Arab 
edition  of  the  Almagest  of  Ptolemy  in  1136,  which  seems  to  have 
been  the  earliest  text-book  among  the  Arabs  that  contained  the 
Arabic  notation,  and  which  it  is  thought  was  instrumental  in  the 
introduction  of  the  system  to  the  Moors  in  Spain.  A  contempo- 
rary of  Gerard,  John  Hispalensis,  a  Jewish  rabbi  converted  to 
Christianity,  translated  several  Arab  and  Moorish  works,  and  also 
wrote  a  treatise  on  algorism,  which  is  said  to  contain  the  earliest 
examples  of  the  extraction  of  the  square  root  of  numbers  by  the 
aid  of  decimal  numbers. 

The  introduction  of  the  Arabic  system  throughout  Europe  pro- 
ceeded slowly.  The  Roman  system  of  calculation  with  the  abacus 
had  been  in  use  many  centuries,  and  it  was  difficult  to  lead  the 
people  to  make  the  transition  from  it  to  the  new  system.  The 
struggle  between  these  two  schools  of  arithmeticians,  the  old 
abacistic  school  and  the  new  algoristic  school,  was  long  and  no 
doubt  often  bitter.  It  was  not  easy  for  the  mathematicians 
and  business  men  who  had  been  brought  up  on  a  system  of  calcu- 
lation with  the  abacus  to  drop  it  and  adopt  the  new  method  of 
computing  with  abstract  symbols. 

One  of  the  most  influential  men  in  bringing  about  the  general 
use  of  the  new  system  was  Leonardo  Fibonacci,  born  at  Pisa  in 
1175.     Educated  in  his  youth   at  Bugia  in  Barbary,  where  his 


ORIGIN    OF    THE    SCIENCE.  25 

father  had  charge  of  the  custom  house,  he  became  acquainted  with 
the  Arabic  system  of  notation  and  with  the  great  work  on  algebra 
by  Al  Khowarazmi.  He  returned  to  Italy  about  1200,  and  in 
1202  composed  a  treatise  on  mathematics  known  as  Liber  abaci, 
in  which  he  explains  the  Arabic  system  of  notation,  and  points 
out  its  great  advantage  over  the  Roman  system.  It  begins  thus  : 
"  The  nine  figures  of  the  Hindoos  are  9,  8,  7,  6,  5,  4,  3,  2,  1. 
With  these  nine  figures  and  with  this  sign,  0,  which  in  Arabic  is 
called  sifr,  any  number  may  be  written."  This  work  had  a  wide 
circulation,  and  practically  introduced  the  use  of  the  Arabic 
system  throughout  Christian  Europe.  It  is  supposed  that  the 
system  was  known  before  this  time  to  the  leading  mathematicians 
who  had  read  the  works  of  Ben  Ezra,  Gerard  and  Hispalensis, 
and  also  by  Christian  merchants  who  had  traded  with  the  Mahom- 
edans,  but  the  wide  reputation  of  Leonardo  gave  a  great  impetus 
to  its  general  adoption. 

The  Arabic  numerals  were  used  at  an  early  day  by  the  astron- 
omers in  composing  calendars,  and  these  calendars  aided  in  dis- 
seminating a  knowledge  of  the  system.  Shortly  after  the  appear- 
ance of  Leonardo's  work,  Alphonso  of  Castile,  in  1252,  published 
some  astronomical  tables  founded  on  observations  made  in  Arabia, 
which  were  computed  by  the  Arabs  and  published  in  the  Arabic 
notation.  A  frequent  and  free  use  of  the  zero  in  the  13th  cen- 
tury is  shown  in  the  tables  for  the  calculation  of  the  tides  at  Lon- 
don and  of  the  duration  of  moonlight.  There  is  an  almanac  pre- 
served in  one  of  the  libraries  of  Cambridge  University  containing  a 
table  of  eclipses  for  the  period  from  1330  to  1348.  This  almanac 
contains  a  brief  explanation  of  the  use  of  numerals  and  the  prin- 
ciples of  the  denary  notation,  indicating  that  at  that  date  the 
system  was  not  generally  understood. 

A  little  tract  in  the  German  language  entitled  De  Algorismo, 
bearing  the  date  of  1390,  explains  with  great  brevity  the  digital 
notation  and  the  elementary  rules  of  arithmetic.  At  the  end  of  a 
short  missal  similar  directions  are  given  in  verse,  which  from  the 
form  of  the  writing  seems  to  belong  to  the  same  period.     The 


26  THE   PHILOSOPHY   OF   ARITHMETIC. 

characters,  of  which  those  in  the  margin  are  fac-similes,  are  in 

both    manuscripts     written     from 

right  to  left,  the  order  which  the      W*/'$*W<*i'£t£CL*')  • 

Arabians  would  naturally  follow. 

The  great  Italian  poet,  Petrarch,  has  the  honor  of  leaving  us 
one  of  the  oldest  authentic  dates  in  the  numeral  characters.  The 
date  is  1375,  written  upon  a  copy  of  St.  Augustine.  The  college 
accounts  in  the  English  universities  were  generally  kept  in 
Roman  numerals  until  the  beginning  of  the  sixteenth  century. 
The  Arabic  characters  were  not  used  in  the  parish  registers  of 
England  before  1G00.  The  oldest  date  met  with  in  Scotland  is 
that  of  1490,  which  occurs  in  the  rent-roll  of  the  Diocese  of  St. 
Andrews.  In  Caxton's  Mirrour  of  the  World,  issued  in  1480, 
there  is  a  wood-cut  of  an  arithmetician  sitting  before  a  table  on 
which  there  are  tablets  with  Hindoo  numerals  upon  them. 

According  to  Fink  the  Roman  symbols  were  generally  used  in 
Germany  with  the  abacus  up  to  the  year  1500.  From  the  15th 
century  on,  these  Hindoo  numerals  appear  more  frequently  in 
Germany  on  monuments  and  in  churches,  but  at  that  time  they 
had  not  become  common  among  the  people.  The  oldest  monu- 
ment in  Germany  with  Arabic  figures  (in  Katherein  near  Trop- 
pau)  is  said  to  date  from  1007,  and  such  monuments  are  found  in 
Pforzheim  (1371)  and  in  Ulm  (1388).  In  the  year  1471  there 
appeared  in  Cologne  a  work  of  Petrarch  with  page  numbers  in 
the  Arabic  figures,  and  in  1482  the  first  German  arithmetic  with 
similar  numbering  was  published  at  Bamberg. 

It  must  have  been  somewhere  from  the  year  1400  to  1450  that 
the  Arabic  system  of  arithmetic  began  to  be  generally  dissemi- 
nated throughout  Europe.  Men  of  science  and  astronomers  had 
become  acquainted  with  the  system  by  the  middle  of  the  13th 
century.  The  trade  of  Europe  during  the  13th  and  14th  centu- 
ries was  mostly  in  Italian  hands,  and  the  advantages  of  the  algor- 
istic  system  led  to  its  adoption  in  Italy  for  mercantile  purposes. 
The  change,  however,  was  not  made  even  among  merchants  with- 
out considerable  opposition  ;  thus  an  edict  was  issued  in  Florence 


ORIGIN    OF    THE    SCIENCE.  27 

in  1299  forbidding  bankers  to  use  the  Arabic  numerals,  and  the 
authorities  of  the  University  of  Padua  in  1348  directed  that  a  list 
should  be  kept  of  books  for  sale  with  prices  marked  4;  non  per 
cifras  sed  per  literas  claras."  Most  merchants  seem  to  have  con- 
tinued to  keep  their  accounts  in  Roman  numerals  until  about  1550, 
and  monasteries  and  colleges  until  about  1650  ;  though  in  both 
cases  it  is  thought  that  the  processes  of  arithmetic  were  performed 
by  the  Arabic  system.  It  was  not  until  the  sixteenth  century 
that  the  Hindoo  position-arithmetic  and  its  notation  first  found 
complete  introduction  among  the  civilized  people  of  the  West. 

The  forms  of  several  of  the  figures  have  undergone  considerable 
change  since  their  first  introduction  into  Europe.  In  the  oldest 
manuscripts  the  figures  4,  5  and  7  are  most  unlike  the  present 
characters.  The  4  consisted  of  a  loop  with  the  ends  pointing  down 
thus  R;  the  5  has  some  likeness  to  the  figure  9,  thus  ^ ,  and  the 
7  is  simply  an  inverted  y,  thus  A-  In  the  dates  used  by  Caxton 
in  the  year  1480,  the  4  has  assumed  its  present  shape,  but  the  5 
and  7  are  still  unlike  the  same  characters  of  to-day.  No  reason  is 
assigned  for  these  changes  ;  they  seem  to  have  been  gradual,  and 
the  result  of  chance  rather  than  of  intention.  The  forms  of  the 
figures  at  different  periods  may  be  seen  in  the  table  given  on  page 
forty-three. 

This  explanation  of  the  introduction  of  the  Arabic  characters 
and  system  of  notation  into  Europe  through  Spain  is  the  one  now 
generally  accepted  as  correct.  M.  Woepcke,  an  excellent  Arabian 
scholar  and  mathematician,  thinks  that  the  Indian  figures  reached 
Europe  through  two  different  channels ;  one  passing  through. 
Enypt  about  the  third  century ;  another  passing  through  Bagdad 
in  the  eighth  century,  and  following  the  track  of  the  victorious 
Islam.  The  first  carried  the  earlier  forms  of  the  Indian  figures 
from  Alexandria  to  Rome,  and  as  far  as  Spain  ;  the  second  carried 
the  later  forms  from  Bagdad  to  the  principal  countries  conquered 
by  the  Kaliffs,  with  the  exception  of  those  where  the  earlier  or 
Gubar  figures  had  already  taken  firm  root.  The  Gubar  figures, 
he  thinks,  were  adopted  by  the  Neo-Pythagoreans,  and  introduced 


28  THE   PHILOSOPHY   OF   ARITHMETIC. 

into  Italy  and  the  Roman  provinces,  Gaul  and  Spain,  as  early  as 
the  sixth  century,  so  that  the  Mohammedans  when  they  reached 
Spain  in  the  eighth  century,  found  these  figures  already  estab- 
lished there,  and  adopted  them.  And  so,  likewise,  when  in  the 
ninth  and  tenth  centuries  the  new  Arabic  treatises  on  arithmetic 
arrived  in  Spain  from  the  East,  they  naturally  adopted  the  more 
perfect  system  of  ciphering  carried  on  without  the  abacus,  and 
kept  the  figures  to  which  they  as  well  as  the  Spaniards  had 
been  accustomed  for  centuries,  and  thus  the  Gubar  figures  were 
retained  by  them.  The  only  change  produced  in  the  ciphering 
of  Europe  by  the  Arabs  was,  he  claims,  the  suppression  of  the 
abacus,  and  the  more  extended  use  of  the  cipher  required  by  the 
new  system  of  reckoning. 

In  the  preparation  of  this  and  the  following  chapter,  I  have  re- 
ceived valuable  assistance  from  Fink's  History  of  Mathematics, 
translated  from  the  German  by  Beman  and  Smith,  and  from  Ball's 
History  of  Mathematics,  both  valuable  works  to  which  the  reader 
is  referred  for  further  information.  I  have  also  received  many 
valuable  suggestions  from  Dr.  David  Eugene  Smith,  Professor  of 
Mathematics  in  Teachers  College,  Columbia  University,  N.  Y. 
The  great  authority  on  the  history  of  mathematics  is  Moritz 
Cantor,  whose  works,  however,  have  not  been  translated  into 
English. 


CHAPTER  III. 

EARLY    WRITERS    ON    ARITHMETIC. 

ANE  of  the  earliest  known  treatises  on  mathematics  is  the 
^  Ahmes  papyrus  of  the  British  Museum.  The  manuscript  was 
written  by  an  Egyptian  scribe  named  Ahmes  sometime  between 
2000  B.  C.  and  1700  B.  C.  The  title  of  the  work  is  "  Directions 
for  Obtaining  the  Knowledge  of  all  Dark  Things."  It  is  believed 
to  be  a  copy,  with  emendations,  of  a  much  older  treatise,  so  that 
it  probably  represents  the  knowledge  of  the  Egyptians  on  arith- 
metic many  centuries  earlier  than  its  own  date.  Two  other 
mathematical  papyri  have  recently  been  found  belonging  to  a 
much  earlier  period  than  that  of  Ahmes,  which  without  entirely 
agreeing  with  the  papyrus  of  Ahmes,  exhibit  many  similarities  to 
it,  especially  in  the  method  of  treating  fractions.  So  that  we  have 
some  knowledge  of  Egyptian  arithmetic  as  early  as  the  twelfth 
dynasty,  or  about  2500  B.  C. 

The  treatise  of  Ahmes  consists  of  the  solution  of  problems  on 
arithmetic  and  geometry  ;  the  answers  are  given,  but  generally 
not  the  processes  by  which  they  were  obtained.  It  deals  with 
both  whole  numbers  and  fractions.  The  treatment  of  fractions  is 
peculiar  in  that  it  is  limited  to  those  having  unity  for  the  numer- 
ator, except  in  the  single  case  of  f.  Fractions  that  cannot  be 
expressed  with  a  unit  numerator  are  represented  by  the  sum  ot 
two  or  more  fractions  whose  numerators  are  each  a  unit ;  thus  for 
|  Ahmes  writes  i  J-g-.  A  fraction  is  designated  by  writing  the 
denominator  with  a  certain  symbol  above  it  to  indicate  its  nature. 
Special  symbols  were  used  for  \,  £,  §  and  £.  Ahmes  treats 
also  of  numerical  equations,  as  when  he  says,  "  heap,  its  seventh, 
its  whole,  it  makes  nineteen  ;"  that  is,  find  a  number  such  that  the 

(29) 


30  THE   PHILOSOPHY   OF   ARITHMETIC. 

sum  of  it  and  one-seventh  of  it  shall  equal  19  ;  the  answer  given 
is  16  -f  i  +  i'  The  word  hau  or  "  heap  "  signifies  the  unknown 
quantity,  or  x.  as  seen  again  in  the  following  :  "  heap,  its  -§, 
its  J,  its  ^,  its  whole,  gives  37  ;  that  is,  ■§#  -f  \x  +  ^x  -\-  x  =  37." 
The  treatise  contains  examples  in  arithmetical  and  geometrical 
progression,  and  employs  the  method  of  "  false  position  "  so 
popular  among  the  Hindoos,  Arabs  and  modern  Europeans. 

The  Greeks  obtained  much  of  their  mathematical  knowledge 
originally  from  the  early  Phoenicians  and  Egyptians.  They  culti- 
vated the  science  of  numbers  to  some  considerable  extent,  but 
failed  to  invent  a  simple  and  convenient  method  of  notation  by 
which  operations  with  numbers  could  be  performed  with  any  de- 
gree of  facility.  Like  many  other  nations  of  antiquity,  they 
depended  upon  the  abacus  in  performing  the  operations  of  the 
fundamental  rules,  though  in  the  time  of  Archimedes  and  Apol- 
lonius  they  could  perform  these  operations  to  some  extent  by 
means  of  their  notation.  The  science  of  arithmetic  with  the 
Greeks  was  speculative  rather  than  practical.  They  did  not  seem 
to  aim  at  the  development  of  skill  in  computation,  but  delighted 
in  investigating  the  properties  of  numbers  and  in  the  discovery  of 
fanciful  analogies  among  them.  It  is  a  matter  of  surprise  that 
while  their  works  on  geometry  have  been  the  models  of  later 
writers  on  that  subject,  the  Greeks  contributed  but  little  of  value 
to  the  science  and  art  of  numbers. 

One  of  the  earliest  Greek  writers  on  mathematics  was  Pytha- 
goras, an  ancient  geometer  who  is  supposed  to  have  lived  from 
about  580  to  about  500  B.  C.  He  brought  from  the  East  a  pas- 
sion for  the  mysterious  properties  of  numbers,  under  tiie  veil  of 
which  he  probably  concealed  some  of  his  secret  and  esoteric  doc- 
trines. He  regarded  numbers  as  of  divine  origin — the  fountain  of 
existence — the  model  and  archetype  of  things — the  essence  of  the 
universe.  He  divided  them  into  classes,  to  each  of  which  were 
assigned  distinct  and  peculiar  properties.  They  were  Even  and 
Odd,  Prime  and  Composite,  Plane  and  Solid,  Triangular,  Square, 
and  Cubical.  Even  numbers  were  regarded  as  feminine  ;  odd 
numbers  as  masculine,  partaking  of  celestial  natures. 


EARLY    WRITERS   ON    ARITHMETIC.  31 

Euclid,  born  about  330  B.  C,  was  one  of  the  early  Greek 
writers  upon  arithmetic.  His  treatise  is  contained  in  the  7th,  8th, 
9th  and  10th  books  of  Euclid's  Elements,  in  which  he  treats  of 
the  theory  of  numbers,  including  prime  and  composite  numbers, 
greatest  common  divisor,  least  common  multiple,  continued  pro- 
portion, geometrical  progressions,  etc.  He  develops  the  theory  of 
prime  numbers,  shows  that  the  number  of  primes  is  infinite, 
unfolds  the  properties  of  odd  and  even  numbers,  and  shows  how 
to  construct  a  perfect  number.  These  books  of  arithmetic  are  not 
included  in  the  common  editions  of  Euclid,  but  are  found  in  an 
edition  by  the  celebrated  Dr.  Barrow.  It  is  supposed  that  Euclid 
was  quite  largely  indebted  to  Thales  and  Pythagoras  for  his 
knowledge  of  the  subject,  though  he  undoubtedly  added  much  to 
the  science  himself.  The  school  at  Alexandria  in  which  he 
taught  was  highly  celebrated,  being  attended  by  the  Egyptian 
monarch  Ptolemy  Lagus.  It  was  this  pupil  to  whom  Euclid, 
upon  being  asked  if  there  was  not  an  easier  method  of  learning 
mathematics,  is  said  to  have  replied,  "  There  is  no  royal  road  to 
geometry ;"  a  statement,  however,  attributed  to  several  other 
mathematicians  of  antiquity. 

Archimedes,  born  about  287  B.  C,  was  one  of  the  most  eminent 
of  the  Greek  mathematicians.  He  is  especially  celebrated  for  the 
discovery  of  the  ratio  of  the  cylinder  to  the  inscribed  sphere,  in 
commemoration  of  which  the  figure  of  a  sphere  inscribed  in  a 
cylinder  was  engraved  upon  his  tomb.  He  wrote  two  papers  on 
arithmetic;  the  object  of  one,  which  is  now  lost,  was  to  explain  a 
convenient  system  of  representing  large  numbers.  The  object  of 
the  second  paper  was  to  show  that  the  method  enabled  one  to 
write  any  number  however  large,  in  which  he  gave  his  celebrated 
illustration  that  the  number  of  grains  of  sand  required  to  fill  the 
universe  is  less  than  1063. 

Eratosthenes,  who  flourished  about  250  years  before  the  Christ- 
ian era,  is  said  to  have  invented  a  method  of  determining  prime 
numbers,  known  as  Eratosthenes'  sieve.  He  is  also  said  to  have 
suggested  the  calendar  now  known   as   the  Julian   Calendar,  in 


32  THE   PHILOSOPHY   OF   ARITHMETIC. 

which  every  fourth  year  contains  366  days.  He  determined  the 
obliquity  of  the  ecliptic,  and  measured  a  degree  on  the  surface  of 
the  earth  which  was  subsequently  found  to  be  too  long  by  about 
nine  miles.  He  also  describes  an  instrument  for  the  duplication 
of  a  cube. 

Nicomachus,  who  is  supposed  to  have  lived  near  the  close  of  the 
first  century  of  the  Christian  era,  wrote  an  arithmetic  which  in 
Latin  translation  remained  for  a  thousand  years  a  standard 
authority  upon  that  subject.  His  special  aim  was  the  investigation 
of  the  properties  of  numbers,  and  particularly  of  ratios.  He  be- 
gins with  the  explanation  of  even,  odd,  prime  and  perfect  num- 
bers ;  then  explains  fractions  in  a  tedious  and  clumsy  manner ; 
then  discusses  polygonal  and  solid  numbers,  and  finally  treats  of 
ratios,  proportion,  and  the  progressions.  He  gives  the  proposition 
that  all  cubical  numbers  are  equal  to  the  sum  of  successive  odd 
numbers;  as  8  =  3  +  5;  27=7  +  9  +  11;  64=13  +  15  +  17+19. 
The  work  was  translated  by  Boethius,  and  was  the  recognized 
text-book  during  the  Middle  Ages. 

Ptolemy  of  Egypt,  who  died  between  125  and  151  A.  D.,  was 
the  author  of  numerous  works  on  mathematics.  His  treatise  on 
astronomy,  called  by  the  Arabs  the  Almagest,  remained  a  standard 
work  on  that  subject  until  the  time  of  Copernicus.  In  this  work 
he  treats  of  trigonometry,  plane  and  spherical,  explains  the 
obliquity  of  the  ecliptic,  uses  3Ty7  as  the  approximate  value  of  *r, 
and  employs  degrees,  minutes,  and  seconds  as  measures  of  angles. 
The  work  exercised  a  strong  influence  in  favor  of  sexagesimal 
arithmetic,  which  uses  the  basis  of  sixty  in  the  representation  of 
numbers. 

Diophantus,  a  mathematician  of  Alexandria,  who  lived  about 
the  beginning  of  the  4th  century,  wrote  a  work  called  Arithmetica. 
It  consists  of  thirteen  books,  only  six  of  which  have  come  down  to 
us.  It  is  really  a  work  on  algebra,  and  before  the  discovery  of 
the  Ahmes  papyrus  was  the  oldest  work  extant  on  that  subject. 
It  treats  of  the  properties  of  numbers,  one  of  his  problems  being 
to  "  divide  a  number,  as  13,  which  is  the  sum  of  two  squares  4  and 


EARLY    WRITERS    ON    ARITHMETIC.  33 

9,  into  two  other  squares,  which  he  finds  to  be  -3^-  and  ^.  It 
presents  solutions  of  simple  and  quadratic  equations,  uses  a  symbol 
for  the  unknown  quantity,  and  shows  that  "  a  number  to  be  sub- 
tracted, multiplied  by  a  number  to  be  subtracted,  gives  a  number 
to  be  added.''  The  work  is  purely  analytic  in  spirit,  and  is  thus 
distinguished  from  the  works  of  other  Greek  writers  like  Euclido 
Diophantus  originated  the  method  of  investigation  known  as  Dio- 
phantine  Analysis. 

Boethius,  born  at  Rome  between  480  and  482,  wrote  an  Arith- 
metic based  on  that  of  Nicomachus.  The  arithmetic  of  Boethius 
was  the  classical  work  of  the  Middle  Ages,  and  became  the  model 
of  several  subsequent  writers  even  down  to  the  fifteenth  century. 
It  was  entirely  theoretical,  treating  of  the  properties  of  numbers, 
particularly  their  ratios,  and  gave  no  rules  of  calculation,  and  wci 
have  no  means  of  telling  whether  the  arithmeticians  of  thii> 
school  reckoned  on  their  fingers,  or  used  an  abacus.  In  the 
manuscript  editions  of  this  work,  current  during  the  11th  century, 
there  is  a  description  of  the  Mensa  Pythagorea,  also  called  the 
abacus  ;  and  mention  is  made  of  nine  figures  which  are  ascribed 
to  the  Pythagoreans  or  Neo  Pythagoreans.  This  passage  is  by 
some  considered  spurious,  and  ascribed  to  a  continuator  of 
Boethius. 

One  of  the  earliest  Hindoo  writers  upon  the  subject  of  mathe- 
matics was  Aryabhatta,  who  was  born  in  Pataliputra  in  476  A.  D. 
His  work,  entitled  Ay-yabhatttyam,  contains  a  number  of  rules  and 
propositions  written  in  verse.  It  consists  of  four  parts,  of  which 
three  are  devoted  to  astronomy  and  the  elements  of  spherical 
trigonometry ;  the  remaining  part  consists  of  thirty-three  rules  in 
arithmetic,  algebra,  and  plane  trigonometry.  The  algebra  shows 
considerable  knowledge  of  the  subject,  but  there  is  no  direct  evi- 
dence that  Aryabhatta  was  acquainted  with  the  modern  method 
of  arithmetic. 

The  next  Hindoo  writer  of  note  is  Brahmngupta,  born  in  598, 
and  was  probably  living  up  to  660.  His  work  entitled  Brahma- 
vphuto  Siddhanta  {i.  e.,  the  improved  system  of  Brahma)  is 
3 


34  THE   PHILOSOPHY   OF   ARITHMETIC. 

written  in  verse,  and  treats  mainly  of  astronomy;  though  two 
chapters  are  devoted  to  arithmetic,  algebra,  and  geometry.  The 
arithmetic  is  entirely  rhetorical  ;  most  of  the  problems  are  worked 
out  by  the  rule  of  three,  and  many  of  them  are  on  the  subject  of 
interest.  His  algebra,  which  is  also  rhetorical,  presents  the 
fundamental  cases  of  arithmetical  progression,  solves  quadratic 
equations,  and  gives  the  method  of  solving  indeterminate  equa- 
tions of  the  second  degree. 

The  first  known  treatise  among  the  Hindoos  which  contains  a 
systematic  exposition  of  the  modern  system  of  arithmetical  nota- 
tion is  that  of  Bhaskara,  born  1114.  This  treatise  was  an 
astronomy,  one  chapter  of  which,  called  Lilawati,  is  an  arithmetic 
written  in  verse,  with  explanatory  notes  in  prose.  After  an  intro- 
ductory preamble  and  colloquy  of  the  gods,  it  begins  with  the 
expression  of  numbers  by  nine  digits  and  the  cipher  or  small  0. 
The  characters  are  similar  to  those  in  present  use,  and  the 
method  of  notation  is  the  same.  It  contains  the  common  rules  of 
arithmetic  and  the  extraction  of  the  square  and  cube  roots.  The 
greater  part  of  the  work  is  taken  up  with  the  discussion  of  the 
"  rule  of  three,"  which  is  used  in  solving  numerous  questions 
chiefly  on  interest  and  exchange. 

Another  chapter  of  Bhaskara's  work  called  Bjita-ganita  (J.  e., 
root  computation)  is  a  treatise  in  algebra.  Abbreviations  and 
initials  are  used  for  symbols  ;  subtraction  is  indicated  by  a  dot, 
addition  by  juxtaposition  merely,  but  no  symbols  are  used  for 
multiplication,  equality,  or  inequality,  these  being  written  out  at 
length.  A  product  is  indicated  by  the  first  syllable  of  the  word 
subjoined  to  the  factors,  between  which  a  dot  is  sometimes  placed. 
In  a  quotient  or  a  fraction,  the  divisor  is  written  under  the  divi- 
dend without  a  line  of  separation.  The  two  sides  of  an  equation 
are  written  one  under  the  other,  confusion  being  prevented  by  the 
recital  in  words  of  all  the  steps  which  accompany  the  operation. 
Various  symbols  for  the  unknown  quantity  are  used,  but  most  of 
them  are  the  initials  of  the  names  of  colors,  and  the  word  color  is 
often  used  as  synonymous  with  unknown   quantity  ;  its  Sanskrit 


EARLY    WRITERS    ON   ARITHMETIC.  35 

equivalent  also  signifies  a  letter,  and  letters  are  sometimes  used 
either  from  the  alphabet  or  from  the  initial  syllables  of  subjects 
of  the  problem.  In  one  or  two  cases  symbols  are  used  for  the 
given  as  well  as  the  unknown  quantities.  The  work  contains  also 
a  treatise  on  trigonometry. 

The  first  Arabic  arithmetic  known  to  us  is  that  of  Al  Kho- 
warazmi, written  about  the  year  830.  It  begins  with  the  words, 
««  Spoken  has  Algoritmi.  Let  us  give  deserved  praise  to  God,  our 
leader  and  defender."  Here  the  name  of  the  author  has  passed 
into  Algoritmi,  from  which  comes  our  modern  word  algorism, 
meaning  the  art  of  computing  in  any  particular  way.  The  work 
treats  of  the  fundamental  rules  by  the  Hindoo  method,  though  the 
forms  of  operation  are  not  so  simple  as  those  now  used.  Al  Kho- 
warazmi  also  wrote  a  work  on  algebra  in  which  the  term  "  algebra," 
al  gebr,  first  occurs.  This  work  holds  an  important  place  in  the 
history  of  mathematics,  as  not  only  subsequent  Arabian,  but 
nearly  all  the  early  mediaeval  works  on  algebra  were  based  on  it. 
It  was  from  the  writings  of  Al  Khowarazmi  that  the  Italians  first 
obtained  their  ideas  of  algebra  and  of  the  modern  method  of  arith- 
metic. This  arithmetic  was  long  known  as  algorism,  or  the  art  of 
Al  Khowarazmi,  in  distinction  from  the  arithmetic  of  Boethius, 
and  this  name  was  retained  until  the  eighteenth  century.  The 
work  had  great  influence  in  introducing  the  Arabic  method  of 
arithmetic  to  the  scholars  and  mathematicians  of  Europe. 

Some  of  the  early  P^uropean  writers  on  arithmetic  were  men- 
tioned in  the  previous  chapter  on  the  origin  and  development  of 
arithmetic.  These  are  Gerbert  of  the  10th  century,  and  Leonardo, 
Ben  Ezra  and  Gerard  of  the  12th  century.  From  the  12th  to  the 
15th  century  there  seem  to  have  been  few  writers  of  note  on  the 
subject  of  mathematics,  the  most  noted  being  Jordanus  of  Ger- 
many in  the  13th  century.  One  of  the  most  distinguished  mathe- 
mnticians  of  the  15th  century  was  Regiomontanus,  who  composed 
a  work  on  trigonometry  in  1464.  This  work  contains  the  earliest 
known  instances  of  the  use  of  letters  to  denote  known  as  well  as 
unknown  quantities. 


36  THE   PHILOSOPHY   OP   ARITHMETIC. 

In  1482  there  appeared  at  Bamberg  a  small  treatise  on  arith- 
metic which  was  attributed  to  Ulrich  Wagner  of  Nuremburg.  It 
was  printed  on  parchment,  and  only  fragments  of  a  single  copy  of 
it  are  now  extant.  In  1483  the  same  Bamberg  publishers  brought 
out  a  second  arithmetic,  printed  on  paper,  and  covering  seventy- 
seven  pages.  The  work  is  anonymous,  but  Ulrich  Wagner  is 
supposed  to  be  its  author.  This  Bamberger  arithmetic  of  1483, 
says  Unger,  bears  no  resemblance  to  previous  Latin  treatises,  but 
aims  especially  at  facility  of  computation  in  mercantile  affairs. 
The  method  of  solution,  as  in  all  the  early  books  on  arithmetic, 
was  that  of  "  the  rule  of  three,"  known  also  as  the  "  merchants' 
rule  "  or  the  "  golden  rule." 

An  arithmetic  by  John  Widmann  was  published  in  Leipzig  in 
1489,  which  is  noted  as  being  the  earliest  book  in  which  the 
symbols  +  and  —  have  been  found,  though  they  had  previously 
appeared  in  a  Vienna  manuscript.  They  were  not  used,  however, 
as  symbols  of  operation,  but  apparently  merely  as  marks  signify- 
ing excess  or  deficiency.  It  is  supposed  by  some  that  they  were 
originally  warehouse  marks  to  indicate  more  or  less  than  the 
normal  weights  of  boxes  or  chests  containing  goods.  In  Widmann's 
book  we  find  equation  of  payments  treated  according  to  the 
methods  still  in  use.  Problems  of  proportional  parts  and  alliga- 
tion were  solved  by  the  use  of  as  many  proportions  as  there  were 
groups  to  be  separated.  The  work  is  obscure  and  deficient  in 
rules  for  operations,  and  abounds  in  fanciful  names  of  topics 
which  Stifel  in  later  years  pronounced  to  be  simply  laughable. 

Lucas  Pacioli,  or  Lucas  di  Borgo,  Jin  Italian  monk,  published 
his  great  work  entitled  Summa  de  Arithmetica,  Geometria,  Propor- 
tion el  Proportionallta  in  Venice  in  1494.  The  work  consists  of 
two  parts,  the  first  dealing  with  arithmetic  and  algebra,  the  second 
with  geometry.  This  is  one  of  the  earliest  printed  treatises  on 
arithmetic  and  algebra,  and  the  earliest  work  presenting  a  system- 
atic exposition  of  algoristic  arithmetic.  It  treats  of  the  four 
fundamental  rules,  and  presents  methods  of  extracting  the  square 
root.     In  its  practical  application  it  deals  largely  with  questions 


EARLY   WRITERS   ON    ARITHMETIC.  37 

relating  to  mercantile  transactions,  including  bills  of  exchange, 
working  out  numerous  examples  in  these  subjects.  It  also  con- 
tains the  first  known  treatise  on  double  entry  book-keeping.  In 
this  work  the  term  "  million  "  and  also  "  nulla  "  or  cero  (zero) 
occurs  for  the  first  time  in  print.  The  work  had  a  wide  influence  in 
the  general  introduction  of  the  new  arithmetic  throughout  Europe. 

Philip  Calandri  published  a  work  on  arithmetic  at  Florence  in 
1491.  It  begins  with  a  picture  of  Pythagoras  teaching,  headed 
"  Pictagoras  Arithmetrice  introductor."  His  notion  of  division 
is  curious.  "When  he  divides  by  8,  he  calls  the  divisor  7,  demand- 
ing, as  it  were,  that  quotient  which,  with  seven  more  like  itself, 
will  make  the  dividend.  He  describes  the  rules  for  fractions,  and 
gives  some  geometrical  and  other  applications. 

Jacob  Kobel,  in  1514,  published,  at  Augsburg,  a  work  on 
arithmetic  in  which  the  Arabic  numerals  are  explained,  but  not 
used.  The  computation  was  by  counters  and  Roman  numerals. 
In  the  frontispiece  is  a  cut  representing  the  mistress  settling 
accounts  with  her  maid-servant  by  an  abacus  with  counters. 

Cuthbert  Tonstall,  in  1522,  published  an  arithmetic  in  Latin 
which  had  great  influence  on  the  development  of  the  science  in 
England.  He  gives  the  multiplication  table  in  the  form  of  a 
square,  and  also  addition,  subtraction,  and  division  tables.  For 
|  of  ^  of  ^  he  writes  j  ^  J ;  and  he  gives  a  clear  explanation  of 
the  multiplication  of  fractions.  De  Morgan  says  this  book  is 
"  decidedly  the  most  classical  which  ever  was  written  on  the  sub- 
ject in  Latin,  both  in  purity  of  style  and  goodness  of  matter." 

Jerome  Cardan  published,  at  Milan,  in  1539,  a  work  entitled 
Practica  Arithmetica.  It  shows,  as  might  have  been  expected 
from  an  Italian  of  that  age,  more  powrer  of  computation  than  the 
French  and  German  writers.  It  contains  a  chapter  on  the 
mystic  properties  of  numbers,  one  use  of  which  is  in  foretelling 
future  events.  These  are  mostly  the  numbers  mentioned  in  the 
Old  and  New  Testaments,  but  not  altogether.  In  another  treat- 
ise, Cardan  expresses  his  opinion  that  it  was  Leonardo  of  Pisa 
who  first  introduced  the  Arabic  numbers  into  Europe. 


38  THE   PHILOSOPHY    OF   ARITHMETIC. 

Robert  Recorde  published  his  celebrated  work  on  arithmetic, 
called  "  The  Grounde  of  Artes,"  about  1540.  It  was  originally- 
dedicated  to  Edward  VI.  The  work  was  subsequently  revised 
and  enlarged  by  John  Dee,  and  published  in  1573,  restoring  the 
original  dedication,  which  had  been  omitted  in  the  edition  pre- 
pared during  the  reign  of  Mary.  This  work  contains  a  number 
of  the  subjects  of  modern  text-books,  including  the  rule  of  three, 
alligation,  fellowship,  false  position,  and  the  method  of  testing 
operations  by  "  casting  out  the  9's."  He  uses  +  and  —  with  the 
explanation,  "  +  whyche  betokeneth  too  muche,  as  this  line,  — , 
plaine  without  a  crosse  line,  betokeneth  too  little."  It  was  sub- 
sequently revised  by  Mellis,  who  added  a  third  part  on  practice 
and  other  things,  and  also  by  Hartwell.  The  last  edition  known 
is  by  Edward  Hatton,  1699,  which  contains  an  additional  book 
called  "  Decimals  made  easie."  It  is  said  to  contain  a  large 
number  of  the  principles  and  problems  of  modern  text-books. 
Recorde  introduced  the  sign  of  equality  (=)  in  a  work  on  algebra, 
published  in  1556.  The  work  was  called  by  the  odd  title,  "  The 
"Whetstone  of  Witte,"  in  which  he  gives  his  reason  for  the  symbol 
in  the  following  quaint  language  :  "  And  to  avoid  the  tedious 
repetition  of  these  words,  I  will  settle,  as  I  doe  often  in  worke  use, 
a  pair  of  parallel  or  Gemowe  lines  of  one  length,  thus,  =,  because 
noe  2  thynges  can  be  more  equalle." 

Michael  Stifel  published,  at  Nuremberg,  in  1544,  his  celebrated 
work  entitled  Arithmetica  Integra.  The  first  two  books  are  on 
the  properties  of  numbers,  on  surds  and  incommensurables, 
learnedly  treated,  and  with  a  full  knowledge  of  what  Euclid  had 
done  on  the  subject.  The  third  book  is  on  algebra,  and  did  much 
for  the  introduction  of  algebra  into  Germany.  Stifel  acknowledges 
his  obligations  to  Adam  Riese,  and  professes  to  have  taken  his 
examples  from  Christopher  Rudolff.  Stifel  was  the  first  to  use  the 
symbols  +  and  —  to  denote  the  operations  of  addition  and  sub- 
traction. He  introduced  also  the  symbol  of  evolution,  p/,  or- 
iginally r,  the  initial  of  radix  or  root,  though  Cantor  says  that 
RudolrF  had  previously  used  it. 


EARLY    WRITERS   ON   ARITHMETIC.  39 

Nicholas  Tartaglia,  an  eminent  Italian  mathematician,  pub- 
lished a  work  on  arithmetic,  vols.  1  and  2  of  which  appeared  in 
1556,  and  vol.  3  in  1560.  The  works  are  verbose,  but  give  a 
clear  account  of  the  various  arithmetical  methods  then  in  use,  and 
present  a  large  number  of  notes  on  the  history  of  arithmetic.  The 
work  on  arithmetic  contains  an  immense  number  of  questions  on 
every  kind  of  problem  which  would  be  likely  to  occur  in  mercan- 
tile arithmetic,  and  attempts  are  made  to  frame  algebraic  formulas 
applicable  to  particular  problems.  It  contains  also  a  large  collec- 
tion of  arithmetical  puzzles  and  questions  of  an  amusing  character, 
among  which  is  found  the  question,  "  What  would  10  be  if  4  were 
6?"  and  the  problem  of  the  three  jealous  husbands  and  their 
wives  who  were  to  cross  a  river  with  a  single  boat  that  would  carry 
only  two  persons.  The  treatise  on  numbers  was  really  an  algebra, 
in  which  are  found  some  interesting  investigations.  Tartaglia  is 
believed  to  be  the  author  of  a  method  of  solving  cubic  equations 
which  Cardan  obtained  from  him  under  a  promise  of  secrecy,  and 
afterward  published  under  his  own  name  in  violation  of  his  promise. 

Simon  Stevinus  published,  at  Leyden,  in  1585,  a  work  which 
was  edited  by  Albert  Girard  in  1634.  This  work  is  character- 
ized by  originality,  accompanied  by  a  great  want  of  the  respect 
for  authority  which  prevailed  in  his  time.  For  example,  great 
names  had  made  the  point  in  jreometry  to  correspond  with  the 
unit  in  arithmetic.  Stevinus  tells  them  that  0,  and  not  1,  is  the 
representative  of  the  point.  "  And  those  who  cannot  see  this," 
he  adds,  "  may  the  Author  of  nature  have  pity  upon  their  un- 
fortunate eyes ;  for  the  fault  is  not  in  the  thing,  but  in  the  sight 
which  we  are  not  able  to  give  them."  A  portion  of  this  work 
contains  "  Les  Tables  d'  Interest "  and  "  La  Disme,"  the  latter 
of  which  exerted  a  great  influence  on  the  introduction  of  decimal 
fractions. 

John  Mellis,  in  1588,  at  London,  published,  "  A  briefe  instruc- 
tion and  manner  how  to  keepe  bookes  of  Accompts  after  the 
order  of  Debitor  and  Creditor,"  etc.  This  is  the  earliest  English 
work  on  book-keeping  by  double  entry.     At  the  end  of  the  book- 


40  THE   PHILOSOPHY    OF   ARITHMETIC. 

keeping  is  a  short  treatise  on  arithmetic.  Mellis  says  :  "  Truly, 
I  am  but  the  renuer  and  reviver  of  an  auncient  old  copie,  printed 
here  In  London  the  14  of  August,  1543.  Then  collected,  pub- 
lished, made  and  set  forth  by  one  Hugh  Oldcastle,  Scholemaster, 
who,  as  appeareth  by  his  treatise  then  taught  Arithmetike  and 
this  booke,  in  Saint  Ollaves  parish  in  Marke  Lane." 

In  1596,  a  work  entitled,  "  The  Pathway  to  Knowledge,"  was 
published  in  London,  which  was  a  translation  from  the  Dutch,  by 
W.  P.  The  translator  gives  the  following  verses,  of  which  he  is 
supposed  to  be  the  author : 

Thirtie  daies  hath  September,  Aprill,  June,  and  November, 
Februarie,  eight  and  twentie  alone ;  all  the  rest  thirtie  and  one. 

Mr.  Davies,  in  his  Key  to  Hutton's  Course,  quotes  the  follow- 
ing from  a  manuscript  of  the  date  of  1570,  or  near  it : 

Multiplication  is  mie  vexation, 

And  Division  is  quite  as  bad, 

The  Golden  Rule  is  mie  stumbling  stule, 

And  Practice  drives  me  mad. 

Cataldi,  successively  Professor  of  Mathematics  at  Florence, 
Perusia  and  Bologna,  published  a  work  on  the  square  root  of 
numbers  at  Bologna,  in  1613.  The  rule  for  the  square  root  is 
exhibited  in  the  modern  form,  and  he  shows  himself  a  most  in- 
trepid calculator.  The  greatest  novelty  of  the  work  is  the  intro- 
duction of  continued  fractions,  then,  it  seems,  for  the  first  time 
presented  to  the  world.  He  reduces  the  square  roots  of  even 
numbers  to  continued  fractions,  and  then  uses  these  fractions  in 
approximation,  but  without  the  aid  of  the  modern  rule  which 
derives  each  approximation  from  the  preceding  two. 

Richard  Witt,  in  1613,  published  a  work  containing  "Arith- 
metical questions  "  on  annuities,  rents,  etc.,  lt  briefly  resolved  by 
means  of  certain  Breviats."  These  Breviats  are  tables,  and  this 
work  is  said  to  be  the  first  English  book  containing  tables  of  com- 
pound interest.  Decimal  fractions  are  really  used.  The  tables 
being  constructed  for  ten  million  pounds,  seven  figures  have  to  be 


EARLY    WRITERS    ON    ARITHMETIC.  41 

cut  off;  and  the  reduction  to  shillings  and  pence,  with  a  temporary 
decimal  separation,  is  introduced  when  wanted.  The  decimal 
separator  used  is  a  vertical  line,  and  the  tables  are  expressly 
stated  to  consist  of  numerators,  with  100...  for  a  denominator. 

John  Napier,  born  1550,  died  1617,  wrote  a  treatise  on  arith- 
metic which  was  published  at  Edinburgh  in  1G17,  after  the 
author's  death.  It  contains  a  description  of  Napier's  rods  with 
applications.  It  is  remarkable  because  it  expressly  attributes  the 
use  of  decimal  fractions  to  Stevinus.  It  also  states  that  Napier 
invented  the  decimal  point.  De  Morgan  says  this  is  not  correct, 
since  1993.273  is  written  19932/7//3///.  Napier  is  illustrious  as 
the  inventor  of  logarithms. 

Robert  Fludd,  in  1617  and  1619,  published  a  work  on  mathe- 
matics at  Oppenheim.  It  contains  two  dedications,  the  first, 
signed  Ego,  homo,  to  his  creator ;  the  second,  on  the  opposite  side 
of  the  leaf,  to  James  I.  of  England,  signed  Robert  Fludd.  The 
first  volume  contains  a  treatise  on  arithmetic  and  algebra.  The 
arithmetic  is  rich  in  the  description  of  numbers,  the  Boethian 
divisions  of  ratios,  the  musical  system,  and  all  that  has  any  con- 
nection with  numerical  mysteries  of  the  sixteenth  century.  The 
algebra  contains  only  four  rules,  referring  for  equations,  etc.,  to 
Stifel  and  Recorde.  The  signs  of  addition  and  subtraction  are  P 
and  M  with  strokes  drawn  through  them.  The  second  volume  is 
strong  upon  the  hidden  theological  force  of  numbers. 

Albert  Girard  published  a  treatise  on  algebra  at  Amsterdam 
in  1629,  which  contains  a  slight  treatise  on  arithmetic.  The 
arithmetic  contains  no  examples  in  division  by  more  than  one 
figure.  On  one  occasion  the  decimal  point  is  used,  though  this 
was  not  the  first  time  it  had  been  employed.  Girard  introduced 
the  parenthesis  in  place  of  the  vinculum,  which  had  been  used  by 
Recorde. 

Wm.  Oughtred's  Clavis  Mathematica,  a  work  on  arithmetic 
and  algebra  of  great  celebrity,  was  first  published  in  1631.  Ife 
retains  the  old  or  scratch  method  of  division  which.  Dr.  Peacock 
observes,  lasted  nearly  to  the  end  of  the  seventeenth  century.   He 


42  THE   PHILOSOPHY   OF   ARITHMETIC. 

does  not  use  the  decimal  point,  but  writes  12.3456  thus :  1213456. 
The  symbol  for  multiplication,  X,  St.  Andrew's  cross,  was  intro- 
duced by  Oughtred.  He  seems  to  have  first  employed  the 
symbol  :  :  to  denote  the  equality  of  ratios.  He  wrote  a  treatise 
on  trigonometry  in  1657,  in  which  abbreviations  for  sine,  cosine, 
etc.,  were  employed. 

Nicholas  Hunt  published,  in  1633,  "  The  Hand-Maid  to  Arith- 
metick  refined."  The  book  is  full  on  weights  and  measures,  and 
commercial  matters  generally.  It  does  not  treat  of  decimal  frac- 
tions, however.  The  author  calls  "  decimall  Arithmeticke, ' 
a  division  of  a  pound  into  10  primes  of  two  shillings  each  ;  each 
shilling  into  six  primes  of  two  pence  each.  It  expresses  the  rules 
in  verse,  of  which  the  following  is  an  example  : 

Adde  thou  upright,  reserving  every  tenne, 
And  write  the  digits  downe  all  with  thy  pen. 
Subtract  the  lesser  from  the  great  noting  the  rest, 
Or  ten  to  borrow  you  are  ever  prest. 
To  pay  what  borrowed  was  think  it  no  paine. 
But  honesty  redounding  to  your  gaine. 

Peter  Herigone,  in  1634,  published  at  Paris  a  work  entitled 
"  Cursus  Mathematici  tomus  secundus."  It  introduces  the  deci- 
mal fractions  of  Stevinus,  having  a  chapter  "  des  nombres  de  la 
dixme."  The  mark  of  the  decimal  is  made  by  marking  the 
place  in  which  the  last  figure  comes.  Thus  when  137  livres  16 
sous  i«  to  be  taken  for  23  years  7  months,  the  product  of  1378' 
and  23583'"  is  found  to  be  32497374"",  or  3249  liv.,  14  sous,  8 
deniers. 

William  Webster  published,  in  1634,  tables  for  simple  and 
compound  interest.  This  work  treats  decimal  arithmetic  as  a 
thing  known.  No  decimal  point  is  recognized,  only  a  partition 
line  to  be  used  on  occasion.  It  contains  the  first  head-rule  for 
turning  a  decimal  fraction  of  a  pound  into  shillings,  pence  and 
farthings.  Many  other  interesting  details  will  be  found  in  the 
works  of  De  Morgan,  Unger,  Fink,  Ball,  Gow  and  Cantor. 


K  © 


EARLY    WRITERS   ON    ARITHMETIC.  43 

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Note.— This  page  of  symbols  is  taken  from  Cajori's  "History  of 
Elementary  Mathematics  "  by  permission  of  the  author  and  publisher. 


CHAPTER  IV. 

ORIGIN    OF    ARITHMETICAL    PROCESSES. 

ONE  of  the  most  interesting  points  connected  with  the  history 
of  arithmetic,  would  be  a  full  and  complete  account  of  the 
genesis  of  the  different  divisions  and  processes  of  the  science. 
This,  however,  is  impossible.  The  origin  of  the  elementary  or 
fundamental  processes  dates  back  before  the  invention  of  printing, 
and  can  never  be  determined.  Some  of  the  principal  facts,  how- 
ever, upon  this  point,  in  addition  to  those  already  given,  will  be 
stated. 

Arithmetical  Language The  notation  of  the  nine  digits 

and  zero,  upon  which  the  science  of  arithmetic  is  based  and 
developed,  originated,  as  we  have  already  shown,  among  the 
Hindoos.  This  notation  was  adopted  by  the  Arabians,  and  be- 
came general  among  Arabic  writers  on  astronomy,  as  well  as 
arithmetic  and  algebra,  about  the  middle  of  the  10th  century. 
From  the  Arabs,  who,  in  the  11th  century,  held  possession  of  the 
southern  provinces  of  Spain,  the  knowledge  was  communicated  to 
the  Spaniards  and  other  nations  of  Europe. 

The  Italians,  from  an  early  period,  adopted  the  method  of  dis- 
tributing the  digits  of  a  number  into  groups  or  periods  of  six,  and 
consequently  proceeding  by  millions.  This  is  the  method  of 
numeration  given  by  Pacioli,  1494.  The  method  of  reckoning  by 
three  places,  as  used  in  this  country  and  on  the  Continent,  seems 
to  have  originated  with  the  Spanish.  In  a  work  on  arithmetic  by 
Juan  de  Ortega,  1536,  we  find  the  following  method  of  numera- 
tion ;  10,  dezena  ;  100,  centena  ;  1000,  miliar  ;  10000,  dezena  de 
miliar;  100000,  centena  de  miliar;  1000000,  cuento.  The  term 
million,  however,  had  not  yet  been  introduced,  and  it  has  not  been 
fully   ascertained   at   what   time   this    introduction   took   place. 

(44) 


ORIGIN    OF   ARITHMETICAL   PROCESSES.  45 

Cantor  says  that  the  term  millione  occurs  the  first  time  in  print  in 
the  summa  dc  arithmetica  of  Paciola.  Bishop  Tonstall,  1522, 
speaks  of  the  term  million  as  in  common  use,  but  rejects  it  as  bar- 
barous, being  used  only  by  the  vulgar. 

Stevinus  divided  numbers  into  periods  of  three  places,  called 
each  period  membres,  aad  distinguished  them  as  le  premier  membre, 
le  seconde  membre,  etc.  Instead  of  million  he  says  mille  mille ;  for 
a  thousand  million  he  uses  mille  mille  mille;  and  for  a  million 
million  he  uses  mille  mille  mille  mille.  It  would  appear  from  the 
practice  of  Stevinus,  and  from  the  observation  of  his  contempor- 
ary, Clavius,  that  the  term  million  was  not  at  this  time  in  general 
use  amongst  mathematicians.  Albert  Girard  divides  numbers 
into  periods  of  six  places,  which  he  terms  premiere  masse*  seconde 
tnasse,  troisieme  masse,  etc.,  the  first  of  which  only  is  divided  into 
periods  of  three  places  each  ;  but  he  does  not  use  the  word  million. 
Ducange  of  Rymer  mentions  the  word  million  in  1514,  and  in 
1540  it  occurs  once  in  the  arithmetic  of  Christopher  Rudolff.  The 
term  was  introduced  into  Recorde's  arithmetic,  1540,  and  subse- 
quently appeared  in  all  succeeding  authors.  It  appears  to  have 
been  admitted  into  German  works  much  later  than  into  the  French 
and  English.  The  terms  billion,  trillion,  etc.,  so  far  as  known, 
appeared  first  in  a  manuscript  work  on  arithmetic  by  Nicolas 
Chuquet,  a  gifted  French  physician  of  Lyons,  and  appear  in 
1520  in  a  printed  work  of  La  Roche. 

Fundamental    Operations The   fundamental    operations 

of  arithmetic  were,  without  doubt,  invented  by  the  Hindoos  at 
a  very  early  period.  The  work  from  which  our  knowledge  of 
Hindoo  arithmetic  has  been  mainly  derived,  is  the  Lilawati  of 
Bhaskara,  who  lived  about  the  middle  of  the  12th  century. 
The  work  is  named  after  the  author's  daughter,  Lilawati,  who, 
it  appeared,  was  destined  to  pass  her  life  unmarried  and  re- 
main without  children.  The  father,  however,  having  ascer- 
tained a  lucky  hour  for  contracting  her  in  marriage,  left  an 
hour-cup  on  a  vessel  of  water,  intending  that  when  the  cup 
should   subside,  the  marriage  should  take  place.     It  happened, 


46  THE    PHILOSOPHY    OF    ARITHMETIC. 

however,  that  the  girl,  from  a  curiosity  natural  to  children, 
looked  into  the  cup  to  see  the  water  coming  in  at  the  hole, 
when,  by  chance,  a  pearl  separated  from  her  bridal  dress,  fell 
into  the  cup,  and  rolling  down  to  the  hole,  stopped  the  influx 
of  water.  When  the  operation  of  the  cup  had  thus  been  de- 
layed, the  father  was  in  consternation ;  and,  examining,  he 
found  that  a  small  pearl  had  stopped  the  flow  of  the  water, 
and  the  long  expected  hour  was  passed.  Thus  disappointed, 
the  father  said  to  his  unfortunate  daughter,  "  I  will  write  a  book 
of  your  name,  which  shall  remain  to  the  latest  times, — for  a  good 
name  is  a  second  life,  and  the  groundwork  of  eternal  existence." 

This  work  frequently  quotes  Brahmagupta,  an  author  who  is 
known  to  have  lived  in  the  early  part  of  the  7th  century,  and 
portions  of  whose  works,  containing  treatises  on  arithmetic  and 
mensuration,  are  still  extant.  Brahmagupta  also  refers  to  an 
earlier  author,  Arabhatta,  who  wrote  an  algebra  and  arithmetic 
as  early  as  the  6th  century,  and  who  is  considered  one  of  the  old- 
est writers  among  the  Hindoos.  In  tracing  the  history  of  the 
operations  of  arithmetic,  we  must  therefore  begin  with  the  Lila- 
wati of  Bhaskara. 

The  fundamental  operations  of  arithmetic,  as  given  in  the 
Lilawati,  are  eight  in  number  ;  namely,  addition,  subtraction, 
multiplication,  division,  square,  square  root,  cube,  cube  root.  To 
the  first  of  these  the  Arabs  added  two,  namely,  dvplation  and 
mediation  or  halving,  considering  them  as  operations  distinct  from 
multiplication  and  division,  in  consequence  of  the  readiness  with 
which  they  were  performed  ;  and  they  appear  as  such  in  many  oi 
the  arithmetical  books  in  the  16th  century. 

Addition. — The  rule  given  in  the  Lilawati  for  addition  is  as 
follows  :  '*  The  sum  of  the  figures,  according  to  their  places,  is 
to  be  taken  in  the  direct  or  inverse  order,"  which  is  interpreted 
to  mean,  "  from  the  first  on  the  right  towards  the  left,  or  from 
the  last  on  the  left  towards  the  right."  In  other  words,  they 
commenced  indifferently  with  the  figures  in  the  highest  or  low- 
est places,   a  practice    which   would   not   lead    to   much   incon 


ORIGIN   OF    ARITHMETICAL    PROCESSES.  47 

lenience  in  their  mode  of  working.     Thus,  to  add  2,  5,  32,  193, 

18,  10,  100,  they  proceed  as  follows: 

Sum  of  the  units,  2,  5,  2,  3,  8,  0,  0,  20 

Sum  of  the  tens,  3,  9,  1,  1,  0,  14 

Sum  of  the  hundreds,  1,  0,  0,  1,  2 

Sum  of  the  sums,  360 

Subtraction. — The  process  of  subtraction  was  also  com- 
menced either  at  the  right  or  the  left,  but  much  more  commonly 
at  the  latter;  and  it  is  remarkable  that  this  method  of  begin- 
r\  ning  to  subtract  at  the  highest  place,  which  is  subject  to 
considerable  inconvenience,  should  have  been  so  general.  It 
is  found  in  Arabic  writers,  in  Maximus  Planudes,  a  Byzantine 
writer  of  about  the  middle  of  the  13th  century,  and  in  many 
European  writers  as  late  as  the  end  of  the  16th  century. 

In  Planudes,  numbers  to  be  added  or  subtracted  are  placed 
one  underneath  another,  as  in  modern  works  on  arithmetic ; 
and  the  sum  or  difference  is  written  above  these  numbers. 
When  a  term  in  the  subtrahend  is  greater  than  the  correspond- 
ing one  in  the  minuend,  a  unit  is  written  beneath  them,  as  in 
the  example  in  the  margin. 

In  performing  the  operation,  3  is  increased     18169  rem. 
by  the  unit  in  the  next  place  to  the  right,  and     54612  min. 
also  5,  8,  4,  and  the  terms  thus  increased  are       , . . .      u  * 
subtracted  from  the  terms  above,  increased  by 
10,  to  find  the  remainder. 

In  other  cases,  the  numbers  are  arranged,  as     06779  rem. 
in  the  margin,  the  digits  3,  0,  0,  2  in  the  minuend     ^99 1 
being  replaced  by  2,  9,  9,   1,  and    then  5  is     0094&    sub 
subtracted  from  4,  4  from  1,  2  from  9,  3  from 
9,  and  2  from  2,  in  order  to  get  the  remainder.     It  is  obvious, 
that  when  such  a  preparation  is  made,  it  is  indifferent  where 
we  commence  the  operation. 

Bishop  Tonstall  attributes  the  invention  of  the  modern 
practice  of  subtraction  to  an  English  arithmetician  of  the  name 
of  Garth.     This  method  he  has  illustrated  with  great  detail, 


2 

9  10  10 

3 

0 

1 

0 

1 

1 

1 

1 

1 

8 

9 

9 

48  THE   PHILOSOPHY   OF   ARITHMETIC. 

and  added,  for  the  assistance  of  the  learner,  a  subtraction  table, 
giving  the  successive  remainders  of  the  nine  digits  when  sub- 
tracted from  the  series  of  natural  numbers  from  11  to  19  inclu- 
sive, the  only  case6  which  can  occur  in  practice. 
In  speaking  of  the  methods  of  preceding  writers, 
he  has  presented  the  example  in  the  margin,  in 
which  it  will  be  seen  that  the  numbers   from 
which    the   subtraction   is   actually   made,   are 
placed  above  the  terms  of  the  minuend. 

In  the  arithmetic  of  Ramus,  which  was  published  in  1584, 
though  written   at   an  earlier  period,  we   find  the   operation 
performed  from  left  to  right,  and  this  method  is  followed 
by  some  other  writers.    Thus,  in  subtracting  345  from       gt 
432  the  terms  to  be  subtracted  and  the  remainder  are    $$jZ 
written  as  in  the  margin.     When  3  is  subtracted  from     $$$ 
4,  the  remainder  should  be  1 ;  but  it  is  replaced  by  zero, 
since  the  next  term  in  the  subtrahend  is  greater  than  the  corres- 
ponding term  of  the  minuend ;  in  the  second  term  the  remainder, 
which  should  be  9,  is  reduced  to  8,   since  5,  the  next  term  of 
the    subtrahend,  is  greater  than  2,  the  term  above  it,  but  the 
last  remainder  7,  is  not  changed. 

Orontius  Fineus,  the  predecessor  of  Ramus  in  the  professor- 
ship of  Mathematics  at  Paris,  in  his  Be  Arithmetica  Practica, 
1555,  subtracts  according  to  the  method  now  used  ;  and  it  is 
difficult  to  account  for  the  adoption  by  Ramus  of  so  inconven- 
ient a  method  as  he  employed,  when  the  method  of  Fineus 
must  have  been  familiar  to  him,  unless  we  attribute  it  to  that 
love  of  singularity  which  led  him  to  aspire  to  the  honor  of 
founding  a  school  of  his  own. 

Multiplication. — The  author  of  Lilawati  has  noticed  six 
different  methods  of  multiplying  numbers,  and  two  others  are 
mentioned  by  his  commentators.  These  may  be  illustrated  by 
their  application  to  the  following  example:  "Beautiful  and 
dear  Lilawati,  whose  eyes  are  like  a  fawn's,  tell  me  what  are 
the  numbers  resulting  from  one  hundred  and  thirty-live  taken 


ORIGIN   OF    ARITHMETICAL    PROCESSES.  49 

into  twelve  ?  If  thou  be  skilled  in  multiplication,  by  whole  or 
by  parts,  whether  by  division  or  separation  of  digits,  tell  me, 
auspicious  woman,  what  is  the  quotient  of  the  product  divided 
by  the  same  multiplier  ?" 

Here  the  multiplicand  is  135,  and  the  multi- 
plier 12;  and  the  first  method,  which  consists  of 
multiplying  the  terms  of  the  multiplicand  suc- 
cessively by  the  multiplier,  is  indicated  in  the 
margin. 

The  second  method,  which  consists  in  sub- 
dividing the  multiplier  into  parts,  as  8  and  4, 
and  severally  multiplying  the  multiplicand  by 
them,  is  also  indicated  in  the  margin.  1620 

The  third  method,  which  con- 
sists in  separating  the  multiplier 
12,  into  its  two  factors,  3  and  4,  and     135  4     20  540  3     120 

multiplying    successively  by  these  y1 

factors,  the  last  product  being  the  — -  1"-" 

result,  is    also  represented  in  the 


r  ii 

L  3  5 
2   12  12 

12  60 
3  6 

135 
135 

16  20 

8  1080 
4  540 

The    fourth    method   consists   in   taking    the 

digits  as  parts,  viz.,  1  and   2,  the  multiplicand     135     135 

being   multiplied    by   them   severally,    and    the       _ 

products  being  added  together  according  to  the  ^™ 

places   of  the   figures,  as  is  represented  in   the  — — 

P        .                    °                         l  1620 
margin. 

The  fifth  method  consists  in  multiplying  the 

multiplicand  by  the  multiplier  less  2,  namely,     135  10  1350 

10,  and  adding  the  result  to  twice  the  multipli-     135  2     270 

cand,  as  may  be  seen  in  the  margin.  1620 

The  sixth  method  consists  in  multiplying  the 

multiplicand  by  the  multiplier  increased  by     135  20  2700 

1  '-?£»  8    1080 

8,  namely,   20,   and  subtracting  8  times  the     LO°  °  1^u 

multiplicand,  as  represented  in  the  margin.  ^"2" 


50 


THE    PHILOSOPHY    OF    ARITHMETIC. 


1 

/  3 

/* 

/2 

/6 

/  ° 

135 
12 

~~ 10 

11 

5 
1 

1620 


The  other  two  methods  are  given  in  the  Commentary  of 
Ganesa.     The  first  of  these,  which  is  \  3  5 

represented  in  the  margin,  appears  to 
have  been  very  popular  in  the  East, 
and  was  adopted  by  the  Arabs,  who 
termed  it  shabacah,  or  net-work,  from 
the  reticulated  appearance  of  the  figure 
which  it  formed,  and  also  by  the  Per-      16  2  0 

sians  under  a  slight  alteration  of  form.  It  is  found  likewise  in 
the  works  of  the  early  Italian  writers  on  algebra,  and  the  same 
principle  may  be  recognized  in  the  process  of  multipli- 
cation by  Napier's  rods. 

The  second  of  these  two  methods  of  multiplica- 
tion, as  represented  in  the  margin,  is  described  in 
full  by  Ganesa.  lie,  however,  considers  this  method 
difficult,  and  not  to  be  learned  by  dull  scholars  with- 
out oral  instruction. 

The  number  and  variety  of  these  methods  would 
seem  to  show  that  the  operation  of  multiplication  was  regarded 
as  difficult,  and  it  is  remarkable  that  the  method  now  used  is  not 
found  amongst  them.  We  find  no  notice  of  the  multiplication 
table  among  either  them  or  the  Arabs.  At  all  events,  it  did 
not  form  a  part  of  their  elementary  system  of  instruction,  a 
circumstance  which  would  account  for  some  of  the  expedients 
which  they  appear  to  have  made  use  of,  for  the  purpose  of 
relieving  the  memory  from  the  labor  of  forming  the  products 
of  the  higher  digits  with  each  other. 

The  Arabs  adopted  most  of  the  Hindoo  methods  of  multi- 
plication, and  added  some  others  of  their  own ;  among  which 
are  some  peculiar  contrivances  for  the  multiplication  of  small 
numbers.  They  may  also  be  considered  as  the  authors  of  the 
method  of  quarter  squares,  or  of  finding  the  product  of  two  num- 
bers by  subtracting  the  square  of  half  their  difference  from  the 
square  of  half  their  sum.  The  Arabs  were  most  probably  the  in- 
ventors of  the  method  of  proof  by  casting  out  9's,  which  is  as  yet 
unknown  to  the  Hindoos  ;  they  called  it  tarazvf  or  the  balance. 


ORIGIN   OF    ARITHMETICAL    PROCESSES.  51 

The  work  of  Planudes  was  chiefly  collected  from  the  Arabic 
writers,  as  appears  from  his  being  acquainted  with  the 
method  of  casting  out  9's.     In  multiplication  he  has     „.~ 
chiefly  followed  the  method  of  multiplying  crosswise  or       35 
Kara  rdv  x'aa/wv,  from  the  figure  x,  which  is  employed  to        x 
connect  the  digits  to  be  multiplied  together.     Thus,  in       24 
multiplying  24  into  35,  we  should  write  the  factors  as  in 
the  margin ;  and  then  multiply  4  into  5  (p^^f),  write  down  0  and 
retain  2  for  the  next  place ;  multiply  4  into  3,  and  3  into  5,  the 
sum  is  22,  which  added  to  2,  makes  24  (ifewzcfef^  write  down  4 
and  retain  2 ;    lastly,  multiply  2  into  3,  add  2,  which  makes 
8  (eKaTovTaSes),  and  the  product  is  840.     He  also  gives  another 
method  which    he    acknowledges  to  be  very   difficult  to  per- 
form with  ink  upon  paper,  but  very  commodious  on  a  board 
strewed  with   sand,  where  the  digits  may  be  readily 
effaced  and  replaced  by  others.     Thus,  taking  the  same     840 
example,  we  multiply  2  into  3,  write  6  above  the  3;     ;,9, 
multiply  2  into  5,  the  result  is  10 ;  add   1  to  6,  and     3  5 
replace  it  by  7,  or  write  7  above  it;  multiply  4  into  3,     2  4 
the  product  is  12;  write  2  above  5,  and  add  1  to  7, 
which  is  replaced  by  8,  or  8  written  above  it ;  lastly,  multiply 
4  into  5,  the  result  is  20;  add  2  to  2,  place  4  above  it  and  after 
it  the  cipher ;  the  last  figures,  or  those  which  remain  without 
accents,  will  express  the  product  required. 

Division. — The  extreme  brevity  with  which  the  rules  of 
division  are  stated  in  the  Lilawati  renders  it  difficult  to 
describe  the  Hindoo  method  of  dividing  numbers.  We  are 
directed  to  abridge  the  dividend  and  divisor  by  an  equal 
number,  whenever  that  is  practicable ;  that  is,  to  divide  them 
both  by  any  common  measure;  thus,  instead  of  dividing  1620 
by  12,  we  may  divide  540  by  4,  or  405  by  3.  We  find,  how- 
ever, in  one  of  the  commentators  on  this  work,  a  description 
of  the  process  of  long  division,  which,  if  exhibited  in  a  scheme, 
would  exactly  agree  with  the  modern  rule 

Italian  Methods. — The  Italians,  who  cultivated  arithmetic 


52  THE    PHILOSOPHY    OF    ARITHMETIC. 

with  so  much  zeal  and  success,  from  a  very  early  period 
adopted  from  their  Oriental  masters  many  of  their  processes 
for  the  multiplication  and  division  of  numbers ;  adding,  how- 
ever, many  of  their  own,  and  particularly  those  which  are 
practiced  at  the  present  time.  In  the  Summa  de  Arithmetica 
of  Lucas  di  Borgo,  we  find  eight  different  methods  of  multi- 
plication, some  of  which  are  designated  by  quaint  and  fanciful 
names.     We  shall  mention  them  in  their  order. 

1.  Multiplicatio :  bericuocoli  e  schacherii.  The  second  of 
these  names  is  derived  from  the  resemblance  of  the  written 
process  to  the  squares  of  a  chess-board ; 

the  first  from  its  resemblance  to  the  check- 
ers  on   a  species  of  sweetmeat  or  cake, 
made  chiefly  from  the  paste  of  bacochi  or 
apricots,  which  were  commonly  used  at 
festivals.     The  process  is  exhibited  in  the 
margin.       This   method   is  presented  by 
Tartaglia  and  later  Italian  writers  with-        17    2   3    6   8 
out  the  squares;  and  it  thus  became  the 
method  which  is  now  universally  used,  and  which  was  adopted 
from  the  beginning  of  the  16th  century  by  all  writers  on  arith- 
metic, nearly  to  the  exclusion  of  every  other  method. 

2.  Castelluccio ;     by  the   little   caslle.      This 
method,    as   indicated  in   the  margin,  uses   the     9876 
upper  number  as  the  multiplier,  and  begins  with       ' 

the  higher  terms.  This  method  was  much  prac-  6J  J?!?00 
ticed  by  the  Florentines,  by  whom  it  was  some-  475^0 
times   called   alV   indielro,    from   the   operation  40734 

beginning  with  the  highest  places,  more  Arabum,  67048164 
according  to  the  statement  of  Pacioli. 

3.  Columna,  o  per  tavoletta ;  by  the  column,  or  by  the  tablets. 
These  were  tables  of  multiplication,  arranged  in  columns,  the 
first  containing  the  squares  of  the  digits,  the  second  the  pro- 
ducts of  2  into  all  digits  above  2  ;  the  third,  of  3  into  all  digits 
above  3 ;  and  so  on,  extending  in  some  cases  as  far  as  the  pro- 


4 

3 

5 

1 

6 

8 

3 

6 

4 

8 

3 

i 

9 

2 

ll 

la 

l« 

|8 

1 

ORIGIN    OF    ARITHMETICAL    PROCESSES. 


53 


20      7 


ducts  of  all  numbers  less  than  100  into  each  other.  Pacioli 
says  that  these  tablets  were  learned  by  the  Florentines,  and 
their  familiarity  with  them  was  considered  by  him  as  a  princi- 
pal cause  of  their  superior  dexterity  in  arithmetical  operatious. 
This  method  is  used  in  multiplying  any  number,  however  large, 
into  another  which  is  within  the  limits  of  the  table.  Thus,  to 
multiply  4G85  by  13,  the  terms  of  the  multiplicand  are  multiplied 
successively  by  13,  and  the  results  formed 
in  the  ordinary  manner. 

4.  Grocetta  sive  casella ;  by  cross  multi- 
plication. This  method  is  said  to  require 
more  mental  exertion  than  any  other,  par- 
ticularly when  many  figures  are  to  be 
combined  together.  Pacioli  expresses  his 
admiration  of  this  method,  and  then  takes 
the  opportunity  of  enlarging  on  the  great  difficulty  of  attaining 
excellence,  whether  in  morals  or  in  science,  without  labor. 

5.  Quadrilatero ;  by  the  square.  This  is  5  4  3 
a  method  which  has  been  characterized  as  5  4  3 
elegant,  and  as  not  requiring  the  operator 
to  attend  to  the  places  of  the  figures  when 
performing  the  multiplications.  The  method 
is  represented  in  the  margin,  and  will  be 
readily  understood. 

6.  Gelosia  sive  graticola;  latticed  multiplication. 
called,"  says  Pacioli,  "because  the  dispo- 
sition of  the  operation  resembles  the  form 
of  a  lattice,  a  term  by  which  we  designate 
the  blinds  or  gratiDgs  which  are  placed  in 
the  windows  of  houses  inhabited  by  ladies 
so  that  they  may  not  easily  be  seen,  as  well 
as  by  other  nuns,  in  which  the  lofty  city  of 
Venice  greatly  abounds."  The  method  will  9  * 
be  readily  understood  by  the  example  given  in  the  margin, 
which  multiplies  987  by  987.     It  is  the  same  as  one  previously 


1  |6|2|9 

2|l|7|2 

1  2  |  7  |  1  |  5 

4 

9    4    8 
It  is  so 


* 

^ 

X 

\  2 

7  \ 

\  4 
6    \ 

\6 

\  1 
8  \ 

X2 
7\ 

9\ 

8      n 


4,  5,  6 
2,  4,  6 

8  10  24 
10  20  30 
1224  36 

30 
GO 
90 

30  GO  90  180 

54  THE    PHILOSOPHY    OF    ARITHMETIC. 

noticed,  which  was  in  common  use  among  the  Hindoos,  Ara- 
bians, and  Persians. 

7.  Ripiego;  multiplication  by  the  unfolding  or  resolution 
of  the  multiplier  into  its  component  factors.  Thus,  to  multiply 
157  by  42,  resolve  42  into  its  ripieghi  or  factors,  6  and  7,  and 
multiply  successively  by  them. 

8.  Scapezzo ;  multiplication  by  cutting  up, 
or  separating  the  multiplier  into  a  number  of 
parts,  which  compose  it  by  addition.  Thus, 
to  multiply  2093  by  17,  we  separate  17  into 
10  and  7,  multiply  by  each,  and  take  the  sum 
of  the  products.  In  some  cases  both  multi- 
plicand and  multiplier  were  separated  into  parts.  Thus,  the 
multiplication  of  15  by  12  was  performed  as  in  the  margin. 

In  another  Italian  arithmetic,  published  in   1567,  by  Pietro 
Cataneo  Sienese,  we  find  the  same  distinctions  preserved,  and 
the  same  names,  or  nearly  so,  attached  to  them  ;  the  method  of 
cross  multiplication  is  expressly  attributed  to  Leonardo       ,.  q 
of  Pisa,  who  derived   it,  in   common    with    Maximus        x 
Planudes,  from  the  Hindoos,  through   the   Arabians.       4  7 
It  is   not   impossible  that   St.  Andrew's  cross,  which       ^ 
is  the    sign   of  multiplication,  was  derived  from    the 
custom  of  uniting  the  numbers  to  be  multiplied  together  by  lines 
which  crossed  each  other,  as  in  the  example  given  in  the  margin. 

Both  Lucas  di  Borgo  and  Tartaglia  mention  other  methods 
of  multiplication  which  were  made  use  of  in  their  time.  An 
extraordinary  passion  seems  to  have  prevailed  in  that  age  for 
the  invention  of  new  forms  of  multiplication,  and  every  pro- 
fessional practitioner  of  arithmetic  considered  it  as  an  important 
triumph  of  his  art  if  he  could  produce  a  figure  more  elegant 
and  more  refined  in  its  composition  and  arrangement  than 
those  which  were  used  by  others.  They  are,  all  of  them,  how- 
ever, characterized  by  Pacioli  as  inconvenient,  at  least  com- 
pared with  those  which  he  had  given  ;  and  Tartaglia  treats  them 
as  trifling  and  superfluous,  such  as  any  one  may  invent  who  is 
acquainted  with  the  2d  proposition  in  the  2d  Book  of  Euclid. 


ORIGIN    OF    ARITHMETICAL    TROCESSES.  55 

The  Hindoos,  as  has  been  stated,  had  no  proper  knowledge 
of  the  multiplication  table,   and  the  Arabs  do  not  appear  to 
have  made  use  of  the  table  of  Pythagoras  as  the  basis  of  their 
arithmetical  education;  the  credit  of  introducing  it,  therefore, 
is  due  to  the  early  Italian  writers  on  the  science,  who  probably 
found  it  in  the  writings  of  Boethius,  and  adopted  it  thence. 
Even  after  the  Italian  arithmeticians  were  familiar  with  this 
table,  many  writers  of  other  countries  considered  it  important 
to  relieve  the  memory  from  the  labor  of  retaining  it  for  the 
products  of   all  digits  exceeding  5,  by  giving  rules  for  their 
formation.     The  principal  rule  for  this  purpose,  called  regula 
ignavi,  or  the  sluggard's  rule,  was  adapted  from  the  Arabians, 
and  is  found  in   Orontius  Fineus,   Recorde,  Laurenberg,  and 
most  other  writers  between  the  middle  of  the  16th 
and  17th  centuries.     The  rule  is  as  follows:   Sub-    7  3  8  2  91 
tract  each  digit  from  10,   and  write  down  the     X    XX 
difference;  multijrfy  these  differences  together,    "4  73  82 
and  add  as  many  tens  to  their  product  as  the    42    5G    72 
first  digit  exceeds  the  second  difference,  or  the 
second  digit  the  first  difference.     The  Arabians  made  use  of 
this  and  other  similar  rules  which  applied  to  numbers  of  two 
places  of   figures,  a  practice  which  may  be  accounted  for  by 
their  very  general  use    of   sexagesimals,  and  the  consequent 
importance  of  being  able  to  form  the  products  which  are  found 
in  a  sexagesimal  table. 

Many  other  expedients  were  proposed  to  relieve  the  mem- 
ory, in  the  process  of  multiplication,  from  the  labor         r .  ,0 
of  carrying  the  tens.     An   interesting  one  is   pre-  43 

sented  by  Laurenberg,  an  author  who  endeavored  ]QQ 

to  elevate  the  character  of   the  common    study  of       1532 
arithmetic  by  collecting  all  his  examples  from  clas-  1^8 

sical  authors,  and  by  making  them  illustrative  of     " 
the  geography,  chronology,  weights  and  measures     221106 
of  antiquity.     It  will   be  understood  from  the  example  given, 
without  explanation. 


56  THE    PHILOSOPHY    OF    ARITHMETIC. 

Division. — Neither  Planudes  nor  the  early  Arabic  writers 
seem  to  have  presented  any  methods  of  dividing  that  merit  the 
special  notice  of  the  writers  on  the  history  of  arithmetic. 
Lucas  di  Borgo  gives  four  distinct  methods  which  we  proceed 
to  explain.  These  methods  had  particular  names,  as  in  mul- 
tiplication. 

1.  Partire  a  regolo,  sometimes  called  also  partire  per  testa 
or  division  by  the  head,  was  used  when  the  divisor  was  a 
single  digit,  or  a  number  of  two  places,  such   as  12, 

13,  etc.,  included  in  the   librettine  or  Italian  tables  6 

of    multiplication.        The    method    will    be   readily     3478 
understood  from  the  example  given.     Di  Borgo  says:       °'9e 
11  This  method  of  division  is  called  by  the  vulgar,  the 
rule,  from  the  similitude  of  the  figure  to  the  carpenter's  rule 
which  is  made  use  of  in  the  making  of  dining-tables,  boxes, 
and  other  articles,  which  rules  are  long  and  narrow." 

2.  Per  ripiego;   which  consists  in  resolving 

the  divisor  into  its  simple  factors,  or  ripieghi.  250047 
It  will  be  readily  understood  from  the  example  g  357 21 
given,  and  be  recognized  as  a  common  method  of  3969 

modern  arithmetics. 

3.  A  danda ;  which  the  author  says  is  thus  called  for  rea- 
sons which  will  be  readily  seen  in  the  opera- 
tion itself,  which  represents  the  division  of 
230265  by  357,  giving  a  quotient  of  645. 
The  process  is  the  same  as  our  common 
method  of   long  division,  only  the  numbers         1606 

are  not   so   conveniently    written.     It  was  1428 

called  a  danda,  or  by  giving,  because  after  1785 

every  subtraction    we  give    or   add    one    or  noo 
more  figures  on  the  right  hand.    The  author, 
however,  prefers  the  next  method. 

4.  Galea  vel  galera  vel  balello ;  so  called  from  tie  process 
resembling  a  galley,  "the  vessel  of  all  others  most  foared'on 


Divisor. 

Proveniens. 

357 

645 

230 

265 

2142 

ORIGIN   OF    ARITHMETICAL    PROCESSES.  57 

the  sea  by  those  who  have  good  knowledge 
of  it ;  the  most  secure  and  swiftest ;  the  most  86 

rapid  and  lightest  of  the  boats  that  pass  on         t jXfti 
the  water.  "     The  method  may  be  illustrated         97o^o399C9 
by  dividing   97535399  by   9876.     We  first  pj$j0 

write  the  dividend,  and   underneath  it  the 
divisor,  and  commence  with  the  second  figure  of  the  dividend, 
since  the  divisor  is  not  contained  in  the  first  four  terms  of  the  divi- 
dend. Multiplying  the  divisor  by  the  first  term 
of  the  quotient,  9  times  9  are  81,  which  sub-         86 
tracted  from  97  leaves  16,  which  is  written         £j/5 
above  97  ;  then  cancel  97  and  9  in  the  divi-       a7?o~qQQ/0Q 
sor;  9  times  8  are  72,  which  taken  from  165,         087^6 
leaves  93 ;  write  9  above   16  and  3  above  5  987 

in   the   dividend,  and   cancel   165,  and  8  in 
divisor;  9  times  7  are  63,  which  subtracted  from  933  leaves 
870;  cancel  933  in  remainder,  and  7  in  divisor;  9  times  6  are 
54,  which  subtracted  from  705  leaves  651 ; 
cancelling  705,  and  6  in  the  divisor,  we  have         }fi 
as  a  remainder  8651399.     For  multiplying        Jffi 
by  the  second  quotient  figure,  we  arrange       -iWia 
the  divisor  as  in  the  margin,  and  proceed  as       ^{mov 
before.      The    complete  operation  is  repre-       97pp05 
sented  by  the  last  work  in  the  margin,  and     Jfi$p).ffl3 
is  so  apparent  that  it  needs  no  further  expla-     ^7p&)$^(9876 
nation.  «ffiP 

Tartaglia  states  that  it  was  the  custom  in  988 

Venice  for  masters  to  propose  to  their  pupils  £) 

as  the  last  proof  of  their  proficiency  in  this 
process  of  division,  examples  which  would  produce  the  com- 
plete form  of  the  galley,  with  its  masts  and  pendant.  The 
last  addition  to  the  work  was  supplied  by  the  scheme  for  the 
proof  of  the  accuracy  of  the  operation  by  casting  out  the 
9's.     Dr.  Peacock   gives  an   example    showing   the    numbers 


58  THE    PHILOSOPHY    OF    ARITHMETIC. 

thus  arranged,  which  is  very  curious,  but  too  long  for  insertion 
here. 

The  same  process  is  illustrated  by  an  example  , , 

from  the  numerous  calculations  by  Regiomon-  3  [34 

tanus,  in  his  tract  on  the  quadrature   of  the        154750 
circle,   written   as  early  as  1464,  though    not        276548 
published  until  1532.     The  question  proposed      ^  J  ??!??' 
is  to  divide  18190735  by  415.     The  divisor  is         41111 
placed    under   the   dividend    and   repeated   at  444 

every  step  backward,  and  all  the  figures  erased      4*3833 
in  succession.     The  quotient,  43833  is  placed 
down  the  side  and  along  the  bottom,  the  remainder  40  being  the 
only  digits  left  on  the  board. 

It  is  amusing  to  observe  the  enthusiastic  admiration  of  Di 
Corgo  for  this  method  of  division.  When  describing  the  pre- 
ceding method  he  seems  impatient,  and  looks  forward  with 
pleasure  to  the  description  of  the  method  a  la  galea,  as  pos- 
sessing a  certain  charm  and  solace,  remarking  that  it  is  a 
noble  thing  to  see  in  any  species  and  scheme  of  numbers,  a 
galley  perfectly  exhibited,  so  as  to  be  able  to  observe  its  mast, 
its  sail,  its  yards  and  its  oars,  launched  in  the  spacious  ocean 
of  arithmetic.  This  method,  we  are  surprised  to  learn, 
appears  to  have  been  preferred  by  nearly  every  writer  on 
arithmetic  as  late  as  the  end  of  the  17th  century.  It  was 
adopted  by  the  Spaniards,  French,  Germans,  and  English;  and 
it  is  the  only  method  which  they  have  thought  necessary  to 
notice.  It  is  found  almost  universally  in  the  works  of  Tonstall, 
Recorde,  Stifel,  Ramus,  Stevinus,  and  Wallis ;  and  it  was 
only  at  the  beginning  of  the  18th  century  that  this  method  of 
division,  called  by  the  English  arithmeticians  the  scratch 
method  of  division,  from  the  scratches  used  in  cancelling  the 
figures,  was  superseded  by  the  method  now  in  common  use, 
which  was  specifically  called  Italian  division,  from  tie  country 
whence  it  was  derived. 


ORIGIN   OF    ARITHMETICAL    PROCESSES.  59 

Recorde  noticed  the    Italian  method  of    33)7890(239^ 
division,  which,  he   says,  "  I  first  learned  66 

190 

of,  and  is  practiced  by  my  ancient  and  espe-  jJJ: 

cial  loving  friend,  Master  Henry  Bridges,         

wherein  not  any  one  figure  is  cancelled  or  297 

defaced.     He  illustrates  the  method  by  an 

example  which  we  subjoin ;  though,  as  before 
stated,  he  preferred  the  scratch  method  of  dividing. 

Powers  and  Roots. — The  author  of  the  Lilawati  has  given 
rules  for  the  formation  of  squares  and  cubes,  as  well 
as   for   the   extraction   of  the   corresponding  roots. 
The  rule  for  the  formation  of  the  square,  which  is       g^ 
very  ingenious,  is  as  follows:  Place  the  square  of      28 
the  last  digit  over  the  number,  and  the  rest  of  the       126 

digits  doubled  and  multiplied  by  the  last  are  to  be     1? 

placed  above  them  respectively ;  then  repeating  the     _???. 
number  with  the  omission  of  the  last  digit,  perform     88209 
the  same  operation.     This  is  illustrated  in  squaring  the  num- 
ber 297. 

In  performing  the  converse  operation,  every  uneven  place  is 
marked  by  a  vertical  line,  and  the  intermediate  digits  by  a 
horizontal  one ;  but  if  the  place  be  even,  it  is  joined 
with  the  contiguous  odd  digit.     It  may  be  illus-      — '  — ' 
trated   by  extracting  the   square   root   of  88209,     8^2^9 
enough  of  the  work  being  indicated  to  show  the     4  8  2  0  9 

nature  of  the  method.     We  subtract  from  the  last  1 

uneven  place,  8,  the  square  4,  and  there  remains     12  2  0  9 
48209,  represented  as  in  the  margin.     Double  the  ~' 

root  2,  making  4,  and  divide  48,  the  number  de- 
noted by  the  next  two  terms,  by  the  result, 
obtaining  9  (10  would  be  too  large),  and  subtracting  9  times  4 
or  36,  we  have  12209.  From  the  uneven  place,  with  the  resi- 
due, 122,  subtract  the  square  of  9,  or  81  ;  the  remainder  is 
4109  Double  9,  giving  18,  and  unite  the  result  with  4,  giving 
58,  and  divide  410  by  it,  and  we  have  7,  and  the  remainder, 


60 


THE   PHILOSOPHY   OF   ARITHMETIC. 


49,  to  which  the  square  of  the  quotient  7,  or  49,  answers  with- 
out a  residue.  The  double  of  the  quotient,  14,  is  put  in  a  line 
with  the  preceding  double  number,  58,  making  594,  the  half 
of  which  is  the  root  sought,  297. 

This  account  of  the  Hindoo  method  of  extracting  square 
root,  is  taken  from  the  commentators  on  the  Lilawati,  and  does 
not  differ  essentially  from  the  method  now  used  ;  and  the  same 
may  be  said  of  the  method  of  extracting  the  cube  root,  the 
principal  difference  from  the  present  method  being  found  in 
their  peculiar  methods  of  multiplying  and  dividing. 

The  method  of  extracting  the  square  root  used  by  the  Ara- 
bians resembled  their   method   of    division  ;    and   it   is  prob- 
able  that  they   are  both   founded   on 
the  Greek  methods  of  performing  these 
operations   with    sexagesimals.       The 
example  given  will  show  the  form  of 
operation.     Vertical  lines  being  drawn 
and   the    numbers   distinguished    into 
periods  of  two  figures,  the  nearest  root 
of  10  is  3,  which  is  placed  both  below 
and  above,  and  its  square,  9,  subtracted  ; 
the  3  is  now  doubled,  and  6  being  writ- 


ten in  the  next  column,  is  contained 
twice  in  17,  or  the  remainder  with  the 
first  figure  of  the  next  period ;  the  2  is 
therefore  set  down  both  above  and 
below,  and  being  multiplied  into  6 
gives  12,  which  is  subtracted  from  17, 
leaving  5  ;  the  square  of  2,  or  4,  is  now 
subtracted  from  55,  and  518,  the  re- 
mainder, with  the  succeeding  figure,  is 
divided  by  G4,  or  the  double  of  32,  giving  8  for  the  quotient; 
then  8  times  64  are  512,  which,  subtracted  from  G18  leaves  6*, 
and  G4  is  exhausted  by  taking  from  it  the  square  of  8.  It  is 
3aid  that  this  mode  was  adopted  from  the  Arabs  by  the  llindoos. 


3 

2 

8 

•                  •                    • 

1 

0 
9 

7 

5 

8 

4 

1 
1 

7 
2 

5 

5 

4 

5 

1 

8 

5 

1 

2 

6 
6 

4 

4 
4 

8 

6 

6 

3 

2 

ORIGIN   OF   ARITHMETICAL   PROCESSES. 


61 


The  earlier  mathematicians  of  Europe  employed  a  similar 
method  of  extracting  the  square  root,  though  perhaps  not  quite  so 
systematic  and  regular.  In  proof  of  the  rule  which  they  followed, 
they  constantly  refer  to  the  4th  proposition  of  the  2d  book  of 
Euclid.    I  will  give  several  examples  illustrating  their  methods 

The  first  is  from  the  arithmetic  of  Pelletier, 
the  first  edition  of  which  was  published  in 
1550.  It  represents  his  method  of  extract- 
ing the  square  root  of  92416,  and  is  so  sim- 
ple it  needs  no  explanation.  It  will  be  seen 
that  the  dots  marking  the  periods  into  which 
the  number  is  separated  are  placed  under  the  number,  instead 
of  above  it  as  is  now  the  custom. 

The  second  example  is  from  the  work  of 
Lucas  di  Borgo,  and  is  in  the  form  of  the 
process  which  was  most  commonly  adopted. 
The  example,  as  will  be  seen,  is  the  extrac- 
tion of  the  square  root  of  99980001.  The 
scheme  will  require  no  explanation,  but  will 
be  readily  understood  by  those  who  are  fam- 
iliar with  the  galley  form  of  division. 


0 
£)2416 

604 
4(304 

2416 


00 

m 

vm 

Mm 

www 

$$gpPp/[(9999 


i 

We  present  another  illustration  taken  from  the  tract,  already 
mentioned,  of  Regiomontanus.  The  question  is  to  find  the 
square  root  of  the  number  5261216896. 
Now  the  nearest  square  to  52  is  49,  leaving 
3  to  be  set  above  the  2,  while  7,  the  root,  is 
placed  in  the  vertical  line ;  then  double  of 
7,  or  14,  being  set  under  the  36,  is  contained 
twice,  and  2  is  accordingly  placed  under  the 
T  ;  but  twice  1  is  2,  which  taken  from  3 
leaves  1,  and  twice  4  are  8,  which  taken 
from  6,  or  16,  leaves  8,  and  extinguishes  the 
1  before  it;  and  twice  2  are  4,  which  taken 
from  1,  or  11,  leaves  T,  and  converts  the  pre- 
ceding 8   into  7.     In  this  way  the  process  advances  till  the 


123 
2465 
1757174 
38796595 
5261216896 
14406 
430 
145 
14 
1 


72534 


62 


THE   PHILOSOPHY    OF    ARITHMETIC. 


figures  become  successively  effaced.  The  root,  72534,  is  placed 
both  at  the  right  hand  side  and  also  immediately  below  the 
work.  The  divisors  do  not  appear  to  be  right,  but  we  do  not 
feel  sufficiently  acquainted  with  the  subject  to  change  them,  and 
do  not  possess  the  original  work  by  which  we  can  verify  them. 
The  method  of  extracting  cube  root  used  by  the  Arabians  and 
Persians,  and  by  them  communicated  to  the 
Hindoos,  resembles  likewise  their  method  f  5 

of  performing  division.  We  will  illus- 
trate it  by  extracting  the  cube  root  of 
91125.  Having  drawn  the  vertical  lines 
as  indicated,  the  several  digits  of  the  num- 
ber are  inscribed  between  them,  and  dots 
set  over  the  first,  fourth,  seventh,  etc., 
reckoning  from  the  right.  The  nearest 
cube  to  91  is  64,  which  is  set  down  and 
subtracted,  leaving  27.  To  obtain  the 
next  term  of  the  root,  3  times  16,  which 
is  3  times  the  square  of  the  root  found,  is 
written    below,    and    being    contained    5 

times  in  271,  the  divisor  is  completed  by 

adding  3  times  the  product  of  4  and  5,  or 

60,  and  then  the  square  of  5,  or  25,  mak- 
ing  in  all   5425,  each  term   of  which   is 

multiplied   by   5,  and  the   products   sub- 
tracted in  succession. 

The  ancient  mode  of  extracting  the  cube  root  practiced  in 

Europe  was  similar  to  the  process 

just  explained,  but  not  so  regular 

and  formal.    The  annexed  example 

is  taken  from  the  Ars  Supputandi 

of  the   famous   Cuthbert  Tonstall, 

Bishop    of  Durham,     the    earliest 

treatise  on  arithmetic  published  in 

England,  and  a  work  of  no  common 

merit.    The  number  250523582464, 


5    4 


5 
5 


4'    7'6' 
3/#  4'0'* 
2W5'2'3'5'8'2/4'6'4/ 


6 


0 


3'4'1'8'7'8'9'8'9'0'4' 
1'0'2'1'5'9' 
1'2'2'5'1'        0' 
4'9'6'    8'2'4'6' 

V 


ORIGIN   OF    ARITHMETICAL    PROCESSES.  63 

whose  root  is  to  be  extracted,  is  placed  above  two  parallel 
lines,  between  which  the  root  6304  is  inserted ;  the  successive 
divisors  and  the  corresponding  remainders  being  written  alter- 
nately below  and  above,  and  the  figures  erased  as  fast  as 
che  operation  advances,  the  operation  of  erasure  being  here 
denoted  by  accents. 

M.  Stifel,  who  usually  sought  to   generalize   the  methods  of 
his  predecessors,  has   considered  the  process  of  extracting  the 
square  root  in  connection  with  those  of  higher  powers.     By 
observing  the  formation  of  the  powers  themselves,  he  discovered 
certain  schemes,  or  pictures  as  he  calls  them,  for  extracting  the 
square,  cube,  biquadrate,  etc.,  roots.     If  we  indicate  the  terms 
of  a  binomial  root  by  a  and  b,  his  scheme  for  the  square  root 
would  consist  of  a-20-b  and  b2  written  under  the  b  to  denote 
addition.     The  meaning  of  the  scheme  is         jjj 
that  in  extracting  the  square  root,  the  first         jjyj8#Z0/(26Ol 
term,  a,  must  be  multiplied  by  20  to  get     2    -  20  -  6. 
the  divisor  from  which  we  determine  the  n  n 

second  term,  b ;  after  which  the  sum  of    2-60-20  1 
the  product  of  a,  20,  and  b,  and  b2  must  1-5201 

be  subtracted  from  the  first  remainder. 

His  method  is  illustrated  by  the  extraction  of  the  square  root 
of  6165201,  as  here  given. 

The  history  of  the  origin  of  these  arithmetical  processes  is 
derived  from  Prof.  Leslie  and  Dr.  Peacock,  much  of  it  having 
been  copied  word  for  word  from  the  originals.  The  origin  of 
methods  in  Fractions,  Decimals,  Bute  of  Three,  Continued 
Fractions,  etc.,  will  be  given  in  connection  with  those  subjects; 
and  such  other  historical  information  as  it  is  thought  will  bo 
of  interest  to  the  reader  will  be  presented  in  its  appropriate 
place.  Occasionally  the  same  fact  is  repeated,  in  order  to  give 
a  completeness  to  the  particular  subject  discussed. 


Part  I. 

THE  NATURE  OF  ARITHMETIC. 


SECTION  I. 
THE  NATURE  OF  NUMBER. 

SECTION  II. 
ARITHMETICAL  LANGUAGE. 

SECTION  III. 
ARITHMETICAL  REASONING. 


SECTION    1. 

THE  NATURE  OF  NUMBER. 


I.     Subject  Matter  of  Arithmetic 


II.     Definition  of  Number 


III.     Classes  of  Numbers. 


IV.     Numerical  Ideas  of  the  Ancient* 


CHAPTER  I. 

NUMBER,   THE   SUBJECT   MATTER   OF   ARITHMETIC. 

NUMBER  was  primarily  a  thought  in  the  mind  of  Deity. 
He  put  forth  His  creative  hand,  and  number  became  a  fact 
of  the  universe.  It  was  projected  everywhere,  in  all  things, 
and  through  all  things.  The  flower  numbered  its  petals,  the 
crystal  counted  its  faces,  the  insect  its  eyes,  the  evening  its 
stars,  and  the  moon,  time's  golden  horologe,  marked  the  months 
and  the  seasons. 

Man  was  created  to  apprehend  the  numerical  idea.  Finding 
it  embodied  in  the  material  world,  he  exclaimed,  with  the  enthu- 
siasm of  Pythagoras,  "  Number  is  the  essence  of  the  universe, 
the  archetype  of  creation."  He  meditated  upon  it  with  enthu- 
siasm, followed  its  combinations,  traced  its  relations,  unfolded 
its  mystic  laws,  and  created  with  it  a  science — the  beautiful 
science  of  Arithmetic.  Let  us  consider  the  origin  and  nature 
of  the  idea  out  of  which  man  has  created  this  science  of  exact 
relations  and  interesting  principles. 

Origin. — The  conception  of  number  begins  with  the  contem- 
plation of  material  objects.  Objects  are  found  in  combinations 
or  collections,  and  the  inquiry,  how  many  of  such  a  collection, 
gives  rise  to  the  idea  of  number.  The  young  mind  looks  out 
upon  nature,  communes  with  its  material  forms,  sees  unity  and 
plurality,  the  one  and  the  many,  all  around  it,  and  awakens  to 
the  numerical  idea.  Strange  law  of  spiritual  development! 
the  material  thing  calls  into  being  the  immaterial  thought. 
The  unitv  and  plurality,  as  it  dwelt  in  the  God-mind  and  was 

(67) 


68  THE   PHILOSOPHY   OF    ARITHMETIC. 

embodied  in  the  material  world,  passes  over  to  the  mind  of 
man,  and  appears  as  an  idea  of  the  immaterial  spirit. 

The  idea  of  definite  riumbers  is  developed  by  a  mental  act 
called  counting.  We  ascertain  the  how-many  of  a  collection, 
by  counting  tbe  objects  in  the  collection.  The  act  of  counting, 
(one,  two,  three,  etc.),  is  the  foundation  of  all  our  knowledge 
of  number.  In  counting,  we  pass  in  succession  from  one 
object  to  another.  Succession  implies  time,  and  is  only  possi- 
ble in  time.  The  idea  of  number,  therefore,  has  its  origin  in 
the  fact  of  time,  and  is  possible  only  in  this  great  fact.  A  brief 
consideration  of  this  relation  will  not  be  uninteresting. 

Time  is  one  of  the  two  great  infinitudes  of  nature.  Space 
and  Time  are  the  conditions  of  all  existence.  Time  enables 
us  to  ask  the  question,  when;  Space,  the  question,  where. 
Space  is  the  condition  of  matter  regarded  as  extended,  and  is 
thus  the  condition  of  extension.  Extension  has  three  dimen- 
sions, length,  breadth,  and  thickness.  The  science  of  extension 
is  geometry.  Space  is  thus  seen  to  be  the  basis  or  condition 
of  the  science  of  geometry. 

Time  is  the  condition  of  events,  as  Space  is  of  objects. 
Every  event  exists  in  Time,  as  every  object  must  exist  in  Space. 
Time  has  somewhat  the  same  relation  to  the  world  of  mind, 
that  Space  has  to  the  world  of  matter.  Matter  extends  in 
Space,  as  mind  protends  in  Time.  This  intimate  relation  of 
Number  and  Time  leads  me  to  present  a  few  thoughts  concern- 
ing the  nature  of  Time,  and  the  development  of  the  idea  of 
Number  from  it. 

Time  is  not  a  mere  abstraction.  It  is  not  a  quality  per- 
ceived in  an  object  and  drawn  away  from  it  by  the  power  of 
abstract  thought,  and  conceived  as  an  abstract  notion.  Neither 
is  it  a  general  idea,  or  a  concept.  We  do  not  first  get  partic- 
ular notions  of  Time,  and  then,  by  putting  these  together,  form 
a  general  idea  of  it.  No  summation  of  particular  times  can 
give  the  grand, unlimited  idea  of  Time  that  the  mind  possesses. 
Indeed,  we  do  not  consider  particular  times  as  examples  of 


NUMBER,    THE   SUBJECT   MATTER   OF   ARITHMETIC.         69 

Time  in  general ;  but  we  conceive  all  particular  times  to  be 
parts  of  a  single  endless  Time.  This  continually  flowing 
and  endless  time  is  what  offers  itself  to  us  when  we  contem- 
plate any  series  of  occurrences.  All  actual  and  possible 
times  exist  as  parts  of  this  original  and  genera*  Time.  There- 
fore, since  all  particular  times  are  considered  as  derivable  from 
time  in  general,  it  is  manifest  that  the  notion  of  time  in  general, 
cannot  be  derived  from  the  notions  of  particular  times. 

Time  is  a  grand  intuition.  It  is  an  idea  which  is  formed  in 
the  mind  when  the  proper  occasion  of  sensible  experience  is 
presented.  Sensible  experience  is  not  the  cause,  but  the  occa- 
sion upon  which  the  mind  conceives  or  originates  this  idea. 
It  is  the  product  of  the  higher  intuitive  power  known  as  the 
Reason.  But  Time  is  not  only  an  idea,  it  is  a  great  reality.  It 
has  a  real  objective  existence,  independent  of  the  mind  which 
conceives  it.  Were  there  no  minds  to  conceive  it,  time  would 
still  exist  as  the  condition  of  events.  Were  all  events  blotted 
out  of  existence,  time  would  remain  an  endless  on-going. 

Time  is  infinite.  No  mind  can  conceive  its  beginning ;  no 
mind  can  conceive  its  end.  All  limited  times  merely  divide, 
but  do  not  terminate  the  extent  of  absolute  time.  In  it  every 
event  begins  and  ends,  while  it  never  begins  and  never  ends. 
It  is,  in  its  very  nature,  like  Him  who  inhabiteth  eternity,  with- 
out beginning  and  without  end. 

Time  gives  rise  to  succession,  as  space  does  to  extension. 
Out  of  succession  grows  the  idea  of  Number,  and  the  science  of 
Number  is  Arithmetic.  Arithmetic,  therefore,  has  somewhat  the 
same  relation  to  time,  that  geometry  has  to  space.  In  view  of 
this  fact,  some  philosophers  have  called  geometry  the  science 
of  space,  and  arithmetic  the  science  of  time.  This  view  of 
arithmetic,  however,  has  not  been  adopted  by  all  writers,  since 
there  are  other  ideas  growing  out  of  time  than  that  of  number. 
Whewell,  in  writing  of  the  Pure  Sciences,  speaks  of  the  three 
great  ideas — Space,  Time,  and  Number  ;  thus  distinguishing 
between  Number  and  Time.     Several  efforts  have  been  made 


70  THE  PHILOSOPHY  OF   ARITHMETIC. 

to  construct  a  science  of  Time ;  the  most  remarkable  is  that  of 
Sir  William  Rowan  Hamilton,  which  resulted  in  the  invention 
of  the  wonderful  Calculus  of  Quaternions. 

Time  is  considered  as  having  but  one  dimension.  In  this 
respect  it  differs  from  Space,  which  has  three  dimensions, 
length,  breadth,  and  thickness.  Time  may  be  regarded  as 
analogous  to  a  line,  but  it  has  no  analogy  to  a  surface  or  a  vol- 
ume. Time  exists  as  a  series  of  instants  which  are  before  and 
after  one  another ;  and  they  have  no  other  relation  than  this 
of  before  and  after.  This  analogy  between  Time  and  a  line 
is  so  close,  that  the  same  terms  are  applied  to  both  ideas,  and 
it  is  difficult  to  say  to  which  they  originally  belonged.  Time 
and  lines  are  called  long  and  short;  we  speak  of  the  beginning 
and  the  end  of  a  line,  of  a  point  of  time,  and  of  the  limits  of  a 
portion  of  duration. 

There  being  nothing  in  Time  which  corresponds  to  more 
than  one  dimension  of  extension,  there  is  nothing  which  bears 
any  analogy  with  figure.  Time  resembles  a  line  extending 
indefinitely  both  ways ;  all  partial  times  are  portions  of  this 
line  ;  and  no  mode  of  conceiving  time  suggests  to  us  a  line 
making  an  angle  with  the  original  line,  or  any  other  combina- 
tion which  might  give  rise  to  figures  of  any  kind.  The  anal- 
ogy between  time  and  space,  which  in  many  circumstances  is 
so  clear,  here  disappears  altogether.  Spaces  of  two  and  of 
three  dimensions,  surfaces  and  volumes,  have  nothing  to  which 
we  can  compare  them  in  the  conceptions  arising  out  of  time. 

The  conception  which  peculiarly  belongs  to  time,  as  figure 
does  to  space,  is  that  of  the  recurrence  of  times  similarly 
marked.  This  may  be  called  rhythm,  using  the  word  in  a 
general  sense.  The  forms  of  such  recurrence  are  noticed  in 
the  versification  of  poetry  and  the  melodies  of  music.  All 
kinds  of  versification,  and  the  still  more  varied  forms  of  recur- 
rence of  notes  of  different  lengths,  which  are  heard  in  all  the 
varied  strains  of  melodies,  are  only  examples  of  such  modifica- 
tions or  configurations,  as  we  may  call  them,  of  time.     They 


NUMBER,  THE    SUBJECT    MATTER   OF   ARITHMETIC.         71 

iQvolve  relations  of  various  portions  of  time,  as  figures  involve 
relations  of  various  portions  of  space.  But  yet  the  analogy 
between  rhythm  and  figure  is  by  no  means  very  close ;  for  in 
rhythm  we  have  relations  of  quantity  alone  in  parts  of  time, 
whereas  in  figure  we  have  relations  not  only  of  quantity,  but 
of  a  kind  altogether  different — namely,  of  positiou.  On  the 
other  hand,  a  repetition  of  similar  elements,  which  does  not 
necessarily  occur  in  figures,  is  quite  essential  in  order  to 
impress  upon  us  that  measured  progress  of  time  of  which  we 
here  speak.  And  thus  the  ideas  of  time  and  space  have  each 
their  peculiar  and  exclusive  relations;  position  and  figure 
belonging  only  to  space,  while  repetition  and  rhythm  are  ap- 
propriate only  to  time. 

One  of  the  simplest  forms  of  recurrence  is  alternation,  as 
we  have  alternate  accented  and  unaccented  syllables  For 
example : 

"  Come  one7,  come  all7,  this  rock7  shall  fly7." 
Or  without  any  subordination,  as  when  we  reckon  numbers, 
and  call  them  in  succession,  odd,  even,  odd,  even,  etc. 

But  the  simplest  of  all  forms  of  recurrence  is  that  which 
has  no  variety,  in  which  a  series  of  units,  each  considered  as 
exactly  similar  to  the  rest,  succeed  one  another;  as  one,  one, 
one,  and  so  on.  In  this  case,  however,  we  are  led  to  consider 
each  unit  with  reference  to  all  that  have  preceded;  and  thus 
the  series  one,  one,  one,  and  so  forth,  becomes  one,  two,  three, 
four,  five,  and  so  on;  a  series  with  which  all  are  familiar, 
and  which  may  be  continued  without  limit.  We  thus  collect 
from  that  repetition  of  which  time  admits,  the  conception  of 
Number. 

This  view  of  the  origin  of  the  idea  of  Number  is  now  accepted 
by  a  large  number  of  thinkers,  but  there  are  those  who  hold  other 
theories.  The  most  objectionable  view  is  that  number  is  a  sense 
perception,  the  absurdity  of  which  is  seen  in  the  fact  that  number 
has  no  color  or  form  or  any  attribute  of  a  percept.  Number  is  not 
a  percept ;  it  is  an  intuition. 


CHAPTER  II. 
DEFINITION  OF  NUMBER. 

THE  idea  of  number  is  so  elementary  that  it  is  difficult  to 
define  it  scientifically.  Various  definitions  have  been  pre- 
sented by  different  writers  upon  the  subject,  though  no  one 
has  hitherto  given  one  which  is,  in  all  respects,  satisfactory. 
The  two  most  celebrated  definitions  are  those  of  Newton  and 
Euclid,  both  of  which  will  be  briefly  considered. 

Newton  defined  number  as  "the  abstract  ratio  of  one  quan- 
tity to  another  quantity  of  the  same  species."  This  definition 
is  philosophical  and  accurate.  It  shows  number  to  be  a  pure 
abstraction  derived  from  a  comparison  of  things.  In  discrete 
quantity,  it  regards  one  of  the  individual  things  as  the  unit  of 
comparison ;  while  in  continuous  quantity  the  unit  is  assumed 
to  be  some  definite  portion  of  the  quantity  considered. 

This  definition  was  no  doubt  primarily  intended  to  apply  to 
extended  quantity,  in  which  there  is  no  natural  unit,  but  in 
which  some  definite  portion  of  the  quantity  is  assumed  as  a 
unit  of  measure,  and  the  quantity  estimated  by  comparing  it 
with  this  unit  as  a  standard.  Such  comparison  gives  rise  to 
three  kinds  of  numbers ;  integral,  fractional,  and  surd  numbers. 
When  the  quantity  measured  contains  the  unit  a  definite 
number  of  times,  the  number  is  integral ;  when  it  is  only  a 
definite  part  of  the  measure,  the  number  is  fractional ;  when 
there  is  no  common  measure  between  the  unit  and  the  quan- 
tity measured,  the  number  is  a  surd  or  radical. 

The  definition  of  Newton,  though  admirable  in  many 
respects,  is  not  suitable  for  popular  use.     It  is  too  abstract  and 

(72) 


DEFINITION   OF   NUMBER.  73 

difficult  to  be  understood  by  young  pupils ;  and  cannot,  there- 
fore, be  recommended  for  our  elementary  text-books.  It  may 
be  said,  also,  that  it  does  not  express  clearly  the  process  of 
thought  by  which  we  attain  the  idea  of  number.  It  is  more 
appropriate  as  applied  to  continuous  than  to  discrete  quantity, 
while  the  idea  of  number  begins  with  discrete  rather  than  con- 
tinuous quantity.  In  this  latter  respect  it  may  possibly  be 
improved  by  changing  the  form  of  expression,  while  retaining 
its  spirit :  thus,  A  number  is  the  relation  of  a  collection  to  the 
single  thing.  This  is  simpler  than  the  original  form,  and  is  in 
many  respects  a  very  satisfactory  definition. 

Euclid  defined  number  to  be  "an  assemblage  or  collection  of 
units  or  things  of  the  same  species."  This  definition,  slightly 
modified,  has  been  generally  adopted  by  mathematicians.  In 
its  original  form  it  excluded  the  number  one,  since  one  thing  is 
not  an  assemblage  or  collection,  and  hence  it  has  been  changed 
to  read — A  number  is  a  unit  or  a  collection  of  units.  This  is 
the  definition  which  is  now  found  in  a  large  number  of  text- 
books. 

This  definition,  however,  is  not  strictly  correct.  A  number 
is  not  precisely  the  same  as  a  collection  of  units,  and  a  collec- 
tion of  units  is  not  necessarily  a  number.  In  other  words, 
there  is  a  difference  between  a  collection  of  things  and  a  num- 
ber of  things.  This  may  be  more  clearly  seen  by  the  use  of 
the  corresponding  verbs.  To  collect  and  to  number  are  two 
different  things.  We  may  collect  without  numbering,  and  we 
may  number  without  collecting ;  I  may  collect  a  number  of 
things,  and  I  may  number  a  collection  of  things.  If  a  basket 
of  apples  were  strewn  over  the  floor  and  I  were  told  to  collect 
them,  I  might  do  so  without  numbering  them  ;  or,  if  told  to 
number  them,  I  might  do  so  without  collecting  them.  In  the 
latter  case  I  would  have  a  number  of  apples  without  having  a 
collection  of  apples,  except  the  mental  collection,  from  which  it 
appears  that  a  number  is  not  precisely  the  same  as  a  collection. 
Number  is  more  definite  than  collection.     A  collection  is  an 


74  THE   PHILOSOPHY   OF   ARITHMETIC. 

indefinite  thing,  numerically  considered  ;  number  is  that  which 
makes  it  definite.  Number  and  collection  are  not,  therefore, 
identical.  Number  is  rather  the  how  many  of  the  collection. 
It  is  thus  seen  that  Euclid's  definition,  as  modified  and  now 
introduced  into  most  of  our  text-books,  is  not  without  scien- 
tific objections.  It  must  be  admitted,  however,  that  there  is 
no  other  one  word  which  so  nearly  expresses  the  idea  of  the 
word  number  as  collection ;  and,  for  ordinary  purposes,  they 
may  be  used  interchangeably.  Thus  we  may  say,  in  analysis, 
we  pass  from  the  collection  to  the  single  thing ;  from  a  number 
to  one.  It  is,  therefore,  regarded  as  the  best  definition  for  the 
ordinary  text-book,  that  has  hitherto  been  presented. 

From  this  discussion  it  will  appear,  as  above  stated,  that  it 
is  difficult  to  present  a  good  definition  of  Number.  This  diffi- 
culty is  due  to  the  fact  that  Number  is  a  simple  term  express- 
ing a  simple  idea,  for  which  we  have  no  other  word  of 
precisely  the  same  signification.  Simple  terms  are  always 
difficult  to  define,  from  the  very  fact  that  they  define  themselves. 
Indeed,  perhaps  there  is  nothing  in  the  way  of  a  definition  of 
number  clearer  than  the  identity — "A  Number  is  a  Number." 
The  following,  though  liable  to  a  verbal  objection,  seems  to  me 
to  come  as  near  the  truth  as  anything  that  has  yet  been  pre- 
sented: A  Number  is  the  how-many  of  a  collection  of  units; 
or,  A  Number  is  how  many  times  a  single  thing  is  reckoned,  or 
is  contained  in  a  collection 

The  first  excludes  the  number  one,  unless,  as  some  writers 
propose,  we  give  a  special  signification  to  collection.  The 
second  provides  for  the  number  one,  but  is  not,  in  other  respects, 
so  satisfactory  as  the  first.  These  definitions  express  pre- 
cisely the  idea  of  a  number,  but  the  use  of  the  expression  how 
many  as  a  noun,  is  not  elegant  in  the  English  language.  The 
simplest  and  most  satisfactory  definition  for  a  text-book  is, 
"  A  Number  is  a  unit  or  a  collection  of  units." 

The  definitions  of  a  number,  as  given  in  some  of  our  text- 
books, are  very  objectionable.     One  author  says :    "  Numbers 


DEFINITION   OF   NUMBER.  75 

are  repetitions  of  units."  This  may  answer  as  a  popular  state- 
ment, but  is  very  far  from  meeting  the  requirements  of  a  sci- 
entific definition.  Another  author  says:  "A  number  is  a 
definite  expression  of  quantity."  So  is  a  triangle  or  a  circle, 
each  of  which  should  be  a  number  if  this  definition  is  correct. 
Another  says:  "A  number  is  an  expression  that  tells  how 
many."  The  two  errors  are,  first,  that  a  number  is  not  an 
expression ;  and,  second,  that  a  number  does  not  tell  anything. 
The  following  definitions  have  also  been  given  by  different 
writers:  "Number  is  a  term  signifying  one  or  more  units;" 
"A  number  is  an  expression  of  one  or  more  things  of  a  kind;" 
11 A  number  is  an  expression  of  quantity  by  a  unit,  or  by  its  repe- 
tition, or  by  its  parts;"  "Number  consists  of  a  repetition  of 
units;"  "A  number  is  either  a  unit  or  composed  of  an  assem- 
blage of  units;"  "A  number  is  a  term  expressing  a  particular 
sameness  of  repetition."  Other  definitions,  equally  incorrect, 
may  be  found  by  leafing  over  text-books  upon  the  subject. 
A  very  simple  definition,  and  especially  suitable  for  a  primary 
text-book  is,  "A  number  is  one  or  more  units."  It  may  bo 
remarked  that  authors  seem  to  be  adopting  the  definition  of 
Euclid,  with  the  modification  presented  above,  so  that  the 
standard  definition  in  our  text-books  is  becoming,  "  A  number 
is  a  unit  or  a  collection  of  units." 

To  give  a  perfect  definition  of  Number  is  exceedingly  diffi- 
cult, if  not  impossible.  Stevinus  defines  it  as  "  that  by  which 
the  quantity  of  anything  is  expressed,"  but  mathematicians 
have  not  adopted  it.  Euler's  definition,  "number  is  nothing 
else  than  the  ratio  of  one  quantity  to  another  quantity  taken 
as  a  unit,"  has  been  highly  commended.  "Number  is  a  defi- 
nite expression  of  quantity,"  has  its  advocates.  "Number  is 
quantity  conceived  as  made  up  of  parts,  and  answers  to  the 
question,  How  many  ?"  has  the  authority  of  a  very  careful 
writer.  The  world,  however,  still  waits  for  a  simple  and  ac- 
curate definition,  which  may  be  generally  adopted. 


CHAPTER  III. 

CLASSES  OF  NUMBERS. 

NUMBERS  have  been  variously  classified  with  respect  to 
different  properties,  or  by  regarding  them  from  different 
points  of  view.  The  fundamental  classes  to  which  attention 
is  here  called,  are  Integers,  Fractions,  and  Denominate  Num- 
bers. These  three  classes  are  practically  and  philosophically 
distinguished,  and  constitute  the  basis  of  three  principal 
divisions  of  the  science  of  arithmetic.  Logically,  the  distinc- 
tion is  not  without  exception,  for  a  Fraction  may  be  denomi- 
nate, and  a  Denominate  Number  may  be  integral;  but  the 
division  is  regarded  as  philosophical,  since  they  are  not  only 
different  in  character,  but  require  distinct  methods  of  treat- 
ment, and  give  rise  to  distinct  rules  and  processes.  The 
philosophical  character  and  relation  of  these  three  classes  of 
numbers,  will  appear  from  the  following  considerations : 

Integers. — The  Unit  is  the  basis  or  beginning  of  numbers. 
A  number  is  a  synthesis  of  units;  it  is  the  how -many  of  a 
collection  of  units.  These  units,  as  they  exist  in  nature,  are 
whole  things,  undivided ;  hence  the  first  numbers  of  which  a 
knowledge  is  acquired,  are  whole  numbers,  that  is,  collections 
of  entire  or  undivided  units.  Such  units,  being  entire,  are 
called  integral  units,  and  the  numbers  composed  of  them  are 
called  integral  numbers,  or  Integers.  An  Integer  is,  therefore, 
a  collection  of  integral  units,  or,  as  popularly  defined,  it  is  a 
whole  number.     It  is  a  product  of  pure  synthesis. 

Fractions. — The  Unit,  as  the  basis  of  arithmetic,  may  be 
multiplied  or  divided.     A  synthesis  of  units,  as  we  have  seen, 

(76) 


CLASSES   OF   NUMBERS.  77 

gives  rise  to  Integers ;  a  division  of  the  unit  gives  rise  to 
Fractions.  Dividing  the  unit  into  a  number  of  equal  parts,  we 
see  that  these  parts  bear  a  definite  relation  to  the  unit  divided, 
and  by  taking  one  or  more  of  these  parts,  we  have  a  Fraction. 
It  is  thus  seen  that  the  conception  of  a  fraction  implies  three 
things :  first,  a  division  of  the  unit ;  second,  a  comparison  of 
the  part  to  the  unit ;  and  third,  a  collection  of  the  fractional 
parts.  In  other  words  it  is  the  product  of  three  operations, 
division,  comparison,  and  collection ;  or,  like  the  logical  nature 
of  the  science  of  arithmetic  itself,  a  fraction  is  a  triune  product, 
consisting  of  analysis,  comparison,  and  synthesis. 

Denominate  Numbers. — The  unit  of  a  simple  integral  num- 
ber exists  in  nature.  A  Denominate  Number  is  a  collection  of 
units  not  found  in  nature;  it  is  a  collection  of  artificial  units 
adopted  to  measure  quantity  of  magnitude.  The  philosophical 
character  of  a  denominate  number  is  indicated  in  the  following 
statement:  Nature,  regarded  as  how  many  and  how  much, 
gives  rise  to  two  distinct  forms  of  quantity ;  quantity  of 
multitude,  and  quantity  of  magnitude.  Quantity  of  multitude 
is  primarily  expressed  by  numbers,  since  it  exists  in  the  form 
of  individuals,  or  units  ;  quantity  of  magnitude  does  not  admit, 
primarily,  of  being  expressed  in  numerical  form.  To  estimate 
quantity  of  magnitude,  we  must  fix  upon  some  definite  part  of 
the  quantity  considered  as  a  unit  of  measure,  by  which  we  can 
give  it  a  numerical  form  of  expression. 

A  Denominate  Number  may,  therefore,  be  defined  as  a 
numerical  expression  of  quantity  of  magnitude.  Or,  since 
the  unit  is  a  measure  by  which  the  quantity  is  estimated,  we 
may  define  it  to  be  a  number  whose  unit  is  a  measure. 
Again,  since  the  unit  is  not  natural  but  artificial,  we  may  de- 
fine it  to  be  a  number  whose  unit  is  artificial.  Either  of  these 
definitions  suffices  to  distinguish  it  from  the  other  two  classes 
of  numbers.  It  differs  from  them  in  respect  of  the  nature  of 
the  quantity  to  which  it  refers,  and  also  in  its  origin  and  com- 
position.    In  the  simple  integral  numbers,  the  units,  as  found 


78  THE   PHILOSOPHY   OF   ARITHMETIC. 

in  nature,  are  collected ;  in  the  denominate  number,  the  unit 
is  assumed,  the  quantity  compared  with  the  unit,  and  the 
result  expressed  numerically.  The  same  kind  of  quantity  may 
be  measured  by  different  units,  bearing  a  definite  relation  to 
each  other,  which  gives  rise  to  a  scale  of  units.  Taking  our 
scales  as  they  now  exist,  we  have  a  series  of  units  definitely 
related  to  each  other,  forming  a  Compound  Number,  which 
does  not  appear  in  the  other  classes  "of  numbers.  This,  how- 
ever, is  rather  incidental  than  essential,  as  it  partially  vanishes 
when  we  apply  the  decimal  scale  to  quantity  of  magnitude,  as 
in  the  metric  system  of  weights  and  measures. 

It  is  thus  seen  that  there  are  three  distinct  classes  of  num- 
bers; and,  since  they  require  different  methods  of  treatment, 
they  will  be  considered  independently.  The  remainder  of  this 
chapter  will  be  devoted  to  the  discussion  of  some  of  the  pecu- 
liarities of  integral  numbers. 

Classes  of  Integers.  —  Simple  Integral  Numbers,  being 
learned  before  Fractions  and  Denominate  Numbers,  are  the 
first  class  to  which  the  term  number  was  applied ;  they  have 
consequently  appropriated  to  themselves  the  almost  exclusive 
use  of  the  word  number.  Thus,  it  is  the  general  custom  to 
speak  of  Numbers,  Fractions,  and  Denominate  Numbers,  appar- 
ently forgetful  that  they  are  all  numbers.  This  custom  being 
so  common,  the  word  Integer  being  somewhat  inconvenient, 
and  some  of  the  properties  which  belong  to  integral  numbers 
applying  also  to  the  other  two  classes,  I  will  also  use  the  word 
number  in  place  of  integral  number  in  considering  this  part 
of  the  subject. 

Numbers  are  of  two  general  classes,  Concrete  and  Abstract. 
A  Concrete  Number  is  a  number  in  which  the  kind  of  unit  is 
named.  An  Abstract  Number  is  a  number  in  which  the  kind 
of  unit  is  not  named.  A  concrete  number  may  also  be  defined 
as  a  number  associated  with  something  which  it  numbers. 
This  is  seen  in  the  etymology  of  the  term,  con  and  cresco,  a 
growing  together.     An  abstract  number  may  also  be  defined 


CLASSES   OF   NUMBERS.  79 

as  a  number  not  associated  with  anything  numbered.  This 
is  indicated  by  the  etymology  of  the  term,  ab  and  traho,  a 
drawing  from.  It  is  not  true,  therefore,  as  has  been  asserted, 
that  "  all  numbers  are  concrete."  Number  is  never  concrete,  in 
the  popular  sense  of  material.  When  I  think  of  four  apples, 
the  apples  are  concrete,  but  the  four  is  purely  numerical  and 
in  no  sense  material.  It  would  be  much  nearer  the  truth  to 
say  that  all  numbers  are  abstract;  for  the  number  itself  is 
always  a  pure  abstraction.  The  distinction  between  an  abstract 
and  a  concrete  number  is  not  a  difference  in  the  numbers  them- 
selves, but  a  distinction  founded  upon  the  fact  of  their  being 
associated  or  not  associated  with  something  numbered. 

This  distinction  is  clearly  seen  in  the  origin  of  the  idea  of 
number.  The  idea  of  number  is  awakened  by  the  contem- 
plation of  material  objects.  The  mind  takes  the  thought  of 
the  how-many,  abstracts  it  from  the  material  things  with  which 
it  was  at  first  associated,  lifts  it  up  into  the  region  of  the  ideal, 
and  conceives  it  as  pure  number.  Though  the  idea  was  pri- 
marily awakened  by  the  objects  of  the  material  world  as  the 
occasion,  yet  so  distinct  is  number  from  matter,  that  if  all 
material  things  were  destroyed,  we  could  still  have  a  science 
of  number  as  complete  as  that  which  now  exists. 

There  is  still  another  method  of  conceiving  the  distinction 
between  concrete  and  abstract  numbers.  All  numbers  are 
composed  of  units.  The  unit  gives  character  and  value  to  the 
number  of  which  it  is  the  basis.  A  number  is  clearly  appre- 
hended only  as  we  have  a  clear  apprehension  of  the  unit :  thus, 
6  pounds  or  6  tons  are  only  clear  and  definite  ideas  to  us  as 
we  have  clear  and  definite  ideas  of  the  units,  pound  and 
ton.  Hence,  also,  the  nature  of  numbers  depends  upon  the 
nature  of  the  units  which  compose  them.  Fundamentally, 
units  are  of  two  classes,  concrete  and  abstract.  A  concrete 
unit  is  some  object  in  nature  or  art,  as,  an  apple,  a  book ;  or 
some  definite  quantity  agreed  upon  to  measure  quantity  of 
magnitude;  as,  a  yard,   a  pound,  etc.      An  abstract    unit  is 


80  THE    PHILOSOPHY    OF    ARITHMETIC. 

merely  one  without  any  reference  to  any  particular  thing.  The 
concrete  unit  is  not  a  number,  it  is  only  one  of  the  things  num- 
bered ;  the  abstract  unit  is  the  number  one.  A  collection  of 
abstract  units  gives  us  an  Abstract  Number;  a  collection  of 
concrete  units  gives  us  what  is  called  a  Concrete  Number. 
An  Abstract  Number  is  thus  merely  a  number  of  abstract 
units  ;  a  Concrete  Number  is  a  number  of  concrete  units.  The 
number  itself  and  the  things  numbered,  considered  together, 
constitute  what  is  called  the  Concrete  Number.  This  is  the 
usual  method  of  conceiving  the  distinction  between  an  abstract 
and  a  concrete  number;  but  it  is  not  as  simple  as  the  one  pre- 
viously presented. 

From  either  method  of  conceiving  the  difference  between 
these  two  classes  of  numbers,  it  will  be  seen  that  the  Concrete 
Number  is  dual  in  its  nature,  consisting  of  two  classes  of  units. 
Thus,  in  the  concrete  number,  four  apples,  the  concrete  unit 
is  one  apple;  while  the  basis  of  the  number  four  itself  is  the 
abstract  unit,  one.  Both  of  these  classes  of  units  must  be 
clearly  apprehended  in  order  to  have  a  clear  and  adequate  idea 
of  any  concrete  number. 


CHAPTER  IV. 

NUMERICAL    IDEAS   OF   THE   ANCIENTS. 

AMONG  the  ancients,  much  time  was  spent  in  discussing 
the  properties  of  numbers.  The  science,  with  them,  was 
mainly  speculative,  abounding  in  fanciful  analogies.  Pythag- 
oras, the  greatest  mathematician  of  his  age,  was  deeply 
imbued  with  this  passion  for  the  mysterious  properties  of 
numbers.  He  regarded  number  as  of  Divine  origin,  the  foun- 
dation of  existence,  the  model  and  archetype  of  things,  the 
essence  of  the  universe. 

Plato  ascribed  the  invention  of  numbers  to  Theuth,  as  may 
be  seen  in  the  following  passage  in  the  Phaedrus :  "  I  have 
heard,  then,  that  at  Naucratis,  in  Egypt,  there  was  one  of  the 
ancient  gods  of  that  country,  to  whom  was  consecrated  the 
bird  which  they  call  Ibis ;  but  the  name  of  the  deity  himself 
was  Theuth.  He  was  the  first  to  invent  numbers,  and  arith- 
metic, and  geometry,  and  astronomy,  and  moreover  draughts 
and  dice,  and  especially  letters."  In  the  Timssus,  he  presents 
the  conception  of  the  relation  of  numbers  to  time,  with  great 
beauty  of  expression.  "Hence,  God  ventured  to  form  a  cer- 
tain movable  image  of  eternity;  and  thus,  while  he  was 
disposing  the  parts  of  the  universe,  he,  out  of  that  eternity 
which  rests  in  unity,  formed  an  eternal  image  on  the  principle 
of  numbers,  and  to  this  we  give  the  appellation  of  Time." 

Aristotle,  in  speaking  of  the  Pythagoreans,  says,  "They 
supposed  the  elements  of  numbers  to  be  the  elements  of  all 
entities,  and  the  whole  heaven  to  be  an  harmony  and  number.'' 
6  (81) 


82  THE   PHILOSOPHY   OP   ARITHMETIC. 

And  again  he  says,  "Plato  affirmed  the  existence  of  numbers 
independent  of  sensibles ;  whereas,  the  Pythagoreans  say  that 
numbers  constitute  the  things  themselves,  and  they  do  not  set 
down  mathematical  entities  as  intermediate  between  these." 

The  views  of  Pythagoras  are  so  curious  and  interesting  that 
they  maybe  stated  somewhat  in  detail.  He  regarded  Numbers 
as  of  Divine  origin,  as  above  stated,  and  divided  them  into 
various  classes,  to  each  of  which  were  assigned  distinct  proper- 
ties. Even  numbers  he  regarded  as  feminine,  and  allied  to 
the  earth ;  odd  numbers  were  supposed  to  be  endued  with 
masculine  virtues,  and  partook  of  the  celestial  nature. 

One,  or  the  monad,  was  held  as  the  most  eminently  sacred, 
as  the  parent  of  scientific  numbers.  Two,  or  the  duad,  was 
viewed  as  the  associate  of  the  monad,  and  the  mother  of  the 
elements,  and  the  recipient  of  all  things  material ;  and  three, 
or  the  triad,  was  regarded  as  perfect,  being  the  first  of  the  mas 
culine  numbers,  comprehending  the  beginning,  middle,  and  end, 
and  hence  fitted  to  regulate  by  its  combinations  the  repetition 
of  prayers  and  libations.  It  was  the  source  of  love  and  sym- 
phony, the  fountain  of  energy  and  intelligence,  the  director  of 
music,  geometry,  and  astronomy.  As  the  monad  represented 
the  Divinity,  or  Creative  Power,  so  the  duad  was  the  image 
of  matter ;  and  the  triad,  resulting  from  their  mutual  con- 
junction, became  the  emblem  of  ideal  forms. 

Four,  or  the  tetrad,  was  the  number  which  Pythagoras 
affected  to  venerate  the  most.  It  is  a  square,  and  contains 
within  itself  all  the  musical  proportions,  and  exhibits  by  sum- 
mation (1  +  2+3+4)  all  the  digits  as  far  as  ten,  the  root  of 
the  universal  scale  of  numeration.  It  marks  the  seasons,  the 
elements,  and  the  successive  ages  of  man ;  and  also  represents 
the  cardinal  virtues,  and  the  opposite  vices.  It  marked  the 
ancient  fourfold  division  of  science  into  arithmetic,  geometry, 
astronomy,  and  music,  which  was  termed  tetractys,  or  quater- 
nion.    Hence,  Dr.  Barrow  explains  the  oath  familiar  to  the 


NUMERICAL   IDEAS   OF  THE  ANCIENTS.  83 

disciples  of  Pythagoras:  "I  swear  by  him  who  communicated 
the  Tetractys."  Five,  or  the  pentad,  being  composed  of  the 
first  male  and  female  numbers,  was  styled  the  number  of  the 
world.  Repeated  in  any  manner  by  an  odd  multiple,  it  always 
reappeared ;  and  it  marked  the  animal  senses  and  the  zones  of 
the  globe. 

Six,  or  the  hexad,  composed  of  the  sum  of  its  several  fac- 
tors (1  +  2+3),  was  reckoned  perfect  and  analogical.  It  was 
likewise  valued  as  indicating  the  faces  of  the  cube,  and  as 
entering  into  the  composition  of  other  important  numbers.  It 
was  deemed  harmonious,  kind,  and  nuptial.  The  third  power 
of  6,  or  216,  was  conceived  to  indicate  the  number  of  years 
that  constitute  the  period  of  metempsychosis. 

Seven,  or  the  heptad,  formed  from  the  junction  of  the  triad 
and  tetrad,  has  been  celebrated  in  every  age.  Being  unpro- 
ductive, it  was  dedicated  to  the  virgin  Minerva,  though  pos- 
sessed of  a  masculine  character.  It  marked  the  series  of  the 
lunar  phases,  the  number  of  the  planets,  and  seemed  to  modify 
and  pervade  all  nature.  It  was  called  the  horn  of  Amalthea, 
and  reckoned  the  guardian  and  director  of  the  universe. 

Eight,  or  the  octad,  being  the  first  cube  that  occurred,  was 
dedicated  to  Cybele,  the  mother  of  the  gods,  whose  image,  in 
the  remotest  times,  was  only  a  cubical  block  of  stone.  From 
its  even  composition,  it  was  termed  Justice,  and  made  to 
signify  the  highest  or  inerratic  sphere. 

Nine,  or  the  ennead,  was  esteemed  as  the  square  of  the 
triad.  It  denotes  the  number  of  the  Muses;  and,  being  the  last 
of  the  series  of  digits,  and  terminating  the  tones  of  music,  it 
was  inscribed  to  Mars.  Sometimes  it  received  the  appellation 
of  Horizon,  because,  like  the  spreading  ocean,  it  seemed  to 
flow  around  the  other  numbers  within  the  decad ;  for  the  same 
reason,  it  was  also  called  Terpsichore,  enlivening  the  productive 
principles  in  the  circle  of  the  dance. 

Ten,  or  the  decad,  from  its  important  office  in  numeration, 
was,  perhaps,  most  celebrated.     Having  completed  the  cycle, 


84  THE   PHILOSOPHY   OP   ARITHMETIC. 

and  begun  a  new  series  of  numbers,  it  was  aptly  called  apo~ 
catastasic,  or  periodic,  and  therefore  dedicated  to  the  double- 
faced  Janus,  the  god  of  the  year.  It  had  likewise  the  epithet 
of  Atlas,  the  unwearied  supporter  of  the  world. 

The  cube  of  the  triad,0T  the  number  twenty-seven,  expressing 
the  time  of  the  moon's  periodic  revolution,  was  supposed  to 
signify  the  power  of  the  lunar  circle.  The  quaternion  of 
celestial  numbers,  one,  three,  five,  and  seven,  joined  to  that  of 
the  terrestrial  numbers,  two,  four,  six,  and  eight,  compose  the 
number  thirty-six,  the  square  of  the  first  perfect  number,  six, 
and  the  symbol  of  the  universe,  distinguished  by  wonderful 
properties. 

In  pursuit  of  these  mystical  relations  and  analogies,  every 
number  became,  as  it  were,  possessed  of  a  property;  and  all 
numbers  possessed  some  relative  analogy  with  each  other  to 
which  a  name  could  be  given.  Numbers  also  became  the  sym- 
bols of  intellectual  and  moral  qualities.  Thus,  perfect  numbers 
compared  with  those  which  are  deficient  or  superabundant,  are 
considered  as  the  images  of  the  virtues,  regarded  as  equally 
remote  from  excess  and  defect,  and  constituting  a  mean  point 
between  them :  thus,  true  courage  is  a  mean  between  audacity 
and  cowardice,  and  liberality  between  profusion  and  avarice. 
In  other  respects,  also,  this  analogy  is  remarkable,  as  perfect 
numbers,  like  virtues,  are  few  in  number,  and  generated  in  a 
constant  order;  while  superabundant  and  deficient  numbers 
are  like  vices,  infinite  in  number,  disposable  in  no  regular 
series,  and  generated  according  to  no  certain  and  invariable 
law. 

The  tracing  of  these  analogies,  accompanied,  as  they  usually 
were,  with  moral  illustrations  of  uncommon  elegance  and 
beauty,  may  be  considered  as  furnishing  a  pleasing,  if  not  a 
useful  exercise  of  the  understanding;  but  such  analogies  were 
often  taken  for  proofs,  and  assumed  as  the  bases  of  the  most 
absurd  and  inconsistent  theories.  Thus  Pythagoras  considered 
"number   as  the  ruler  of  forms  and  ideas,  and  the  cause  of 


NUMERICAL   IDEAS   OF   THE    ANCIENTS.  85 

gods  and  daemons;"  and  again  that  'Ho  the  most  ancient  and 
all-powerful  creating  Deity,  number  was  the  canon,  the  efficient 
reason,  the  intellect  also,  and  the  most  undeviating  of  the 
composition  and  generation  of  all  things."  Philolaus  declared 
"that  number  was  the  governing  and  self-begotten  bond  of 
the  eternal  permanency  of  mundane  natures."  Another  said, 
"that  number  was  the  judicial  instrument  of  the  Maker  of 
the  universe,  and  the  first  paradigm  of  mundane  fabrication." 

It  appears  to  have  been  a  favorite  practice  with  the  Greeks 
of  the  latter  ages  to  form  words  in  which  the  sum  of  the  num- 
bers expressed  by  their  component  letters,  should  be  equal  to 
some  remarkable  number ;  of  this  kind  were  the  words  appacai 
and  appacaga,  the  letters  in  which  express  numbers,  which  added 
together,  are  equal  to  365  and  366,  the  number  of  days  in  the 
common  and  bissextile  years  respectively;  and  it  was  also 
remarked  that  the  word  veilog  possessed  the  same  property  as 
the  first  of  these  words.  Words  in  which  the  sums  of  the 
numbers  expressed  by  the  letters  were  equal,  were  called 
w6/iara  la&^ta;  and  we  have  an  example  in  the  Greek  anthol- 
ogy, where  a  poet,  wishing  to  express  his  dislike  to  a  fellow  of 
the  name  of  Aafcayopac,  says,  that  having  heard  that  his  name 
was  equivalent  in  numeral  value  to  Aoifibc,  a  pestilence,  he  pro- 
ceeded to  weigh  them  in  a  balance,  when  the  latter  was  found 
to  be  the  lighter. 

Observations  like  these,  however  trifling,  are  not  without 
their  portion  of  curiosity ;  but  the  same  indulgence  cannot  be 
shown  to  the  absurdities  of  those  Pythagorean  philosophers, 
who,  among  other  extraordinary  powers  which  they  attributed 
to  numbers,  maintained  that,  of  two  combatants,  the  one  would 
conquer,  the  characters  of  whose  name  expressed  the  larger 
sum.  It  was  upon  this  principle  that  they  explained  the  rela- 
tive prowess  and  fate  of  the  heroes  in  Homer,  iLarpoicloc,  Enrop, 
and  AxtXteve,  the  sums  of  the  numbers  in  whose  names  are  871, 
1225,  and  1216  respectively. 

This  very  singular  superstition  continued  in  force  as  late  as 


86  THE   PHILOSOPHY   OF   ARITHMETIC. 

the  sixteenth  century,  and  was  transferred  from  the  Greek  to 
the  Roman  numeral  letters,  I,  TJ  or  V,  X,  L,  C,  D,  and  M, 
which  correspond  to  the  numbers  1,  5,  10,  50,  100,  500,  and 
1000 ;  thus  the  numeral  power  of  the  name  of  Maurice  (Mau- 
ritius) of  Saxony,  was  considered  as  an  index  of  his  success 
against  Charles  Y.  It  was  the  fashion,  also,  to  select  or  form 
memorial  sentences  or  verses  to  commemorate  remarkable 
dates.  Thus  the  year  of  the  Reformation  (151*7)  was  found  tc 
be  expressed  by  the  numeral  letters  of  this  verse  of  the  Te 
Deum,  Tibi  cherubin  et  seraphin  incessabili  voce  proclamant, 
in  which  there  is  one  M,  four  C's,  two  L's,  two  TJ's  or  Y's,  and 
seven  l's. 

The  Chinese,  also,  are  distinguished  for  their  arithmetical 
fancies.  They  regarded  even  numbers  as  terrestrial,  and  par- 
taking of  the  feminine  principle  Yang;  while  odd  numbers 
were  regarded  as  of  celestial  extraction,  and  endued  with  the 
masculine  principle  F.  Even  numbers  were  represented  by 
small  black  circles ;  odd  numbers  by  small  white  ones,  vari- 
ously disposed  and  connected  by  straight  lines.  Thirty,  the 
sum  of  the  five  even  numbers,  2,  4,  6,  8,  and  10,  was  called 
the  number  of  the  Earth ;  twenty-five,  the  sum  of  the  odd 
numbers,  1,  3,  5,  7,  9,  and  also  the  square  of  five,  was  called 

the  number  of  Heaven. 

o— o— o— o— o  • 

\  /  / 

o-o-o-o  • 


The  nine  digits  were  grouped   y#\ 
in  two  ways  called  Lo-chou  and    \^/ 
Ho-tou.      The   former  expres- 
sion signifies  the  Book  of  the  o  \  /  o 
River  Lo.  or  what  the  Great  Yu  1                    \/                   ,o 
saw  delineated  on  the  back  of  J                     /   \                  |    o 
the  mysterious  tortoise  which                  0'          ^0             xo 
rose  out  of  that  river.     It  may  o 
be  represented  as  follows :  Nine          ♦/^x#                            • 
was  the    head,   one    the  tail,      j^   if                         •.     •. 
three   and  seven   its  left   and   Vr                                   *s  / 
right  shoulders,  four  and  two 


NUMERICAL   IDEAS   OF    THE   ANCIENTS.  87 

its  fore  feet,  eight  and  six  its  hind  feet.  The  number  Jive, 
which  represented  the  heart,  being  the  square  root  of  twenty 
five,  was  also  the  emblem  of  Heaven.  It  will  be  noticed  that 
this  group  of  numbers  is  the  common  magic  square  of  nine 
digits,  each  row  of  which  amounts  to  fifteen. 

The  Ho-tou  was  what  the  Emperor  Fou-hi  observed  on  the 
body  of  the  horse-dragon  which  he  saw  spring  out  of  the 
river  Ho.  It  consists  of  the 
first  nine  numbers  arranged  in 
the  form  of  across.  The  central 

number   was  ten,  which,  it  is  £ 

remarked  by  the  commentators,    4  #        •        •         #6 

terminates  all  the  operations  on    f  o     #        •        •        •    •  V 
numbers.    Other  facts  equally    J  Y  •        •        •         ^  A 

curious  will    be  found  in  the    *  O        o  9 

literature  of   other  nations,  a    f  O  9 

full  collection  of  which  would  o        o 

make   an   interesting   volume. 

For  the  facts  here  presented,  •  •  •  •  • 

and  the  manner  in  which  they  are  stated,  I  am  indebted  to 
Leslie. 

This  passion  for  discovering  the  mystical  properties  of  num- 
bers descended  from  the  ancients  to  the  moderns,  and  numer- 
ous works  have  been  written  for  the  purpose  of  explaining 
them.  Petrus  Bungus,  in  1618,  wrote  a  work  on  the  mysteries 
of  numbers,  extending  to  seven  hundred  quarto  pages.  He 
illustrates  all  the  properties  of  numbers,  whether  mathemat- 
ical, metaphysical,  or  theological ;  and  not  content  with  col- 
lecting all  the  observations  of  the  Pythagoreans  concerning 
them,  he  has  referred  to  every  passage  in  the  Bible  in  which 
numbers  are  mentioned,  incorporating,  in  a  certain  sense,  the 
whole  system  of  Christian  and  Pagan  theology.  He  holds  that 
the  number  11,  which  transgresses  the  decad,  denotes  the 
wicked  who  transgress  the  Decalogue,  whilst  12,  the  numbe' 
of  the  apostles,  is  the  proper  symbol  of  the  good  and  the  just. 


88  THE    PHILOSOPHY   OF    ARITHMETIC. 

The  number,  however,  upon  which,  above  all  others,  he  haa 
dilated  with  peculiar  industry  and  satisfaction,  is  666,  the  num- 
ber of  the  beast  in  Revelation,  the  symbol  of  Antichrist ;  and 
he  seems  particularly  anxious  to  reduce  the  name  of  Martin 
Luther  to  a  form  which  may  express  this  formidable  number. 
It  may  also  be  remarked  that  Luther  interpreted  this  number 
to  apply  to  the  duration  of  Popery,  and  also  that  his  friend  and 
disciple,  Stifel,  the  most  acute  and  original  of  the  early  math- 
ematicians of  Germany,  appears  to  have  been  seduced  by  these 
absurd  speculations. 

The  numbers  3  and  7  were  the  subject  of  particular  specula- 
tion with  the  writers  of  that  age ;  and  every  department  of 
nature,  science,  literature,  and  art,  was  ransacked  for  the  pur- 
pose of  discovering  ternary  and  septenary  combinations.  The 
excellent  old  monk,  Pacioli,  the  author  of  an  early  printed 
treatise  on  arithmetic,  has  enlarged  upon  the  first  of  these 
numbers  in  a  manner  which  is  rather  amusing,  from  the  quaint 
and  incongruous  mixture  of  the  objects  which  he  has  selected  for 
illustration.  "  There  are  three  principal  sins,"  says  he,  "  avarice, 
luxury,  and  pride  ;  three  sorts  of  satisfaction  for  sin, — fasting, 
almsgiving,  and  prayer;  three  persons  offended  by  sin, — God, 
the  sinner  himself,  and  his  neighbor;  three  witnesses  in 
heaven, — the  Father,  the  Word,  and  the  Holy  Spirit;  three 
degrees  of  penitence, — contrition,  confession,  and  satisfaction, 
which  Dante  has  represented  as  the  three  steps  of  the  ladder 
that  leads  to  Purgatory,  the  first  marble,  the  second  black  and 
rugged  stone,  the  third  red  porphyry.  There  are  three  Furies 
in  the  infernal  regions;  three  Fates, — Atropos,  Lachesis,  and 
Clotho;  three  theological  virtues, — faith,  hope,  and  charity; 
three  enemies  of  the  soul, — the  world,  the  flesh,  and  the  devil ; 
three  vows  of  the  Minorite  Friars, — poverty,  obedience  and 
chastity ;  three  ways  of  committing  sin, — with  the  heart,  the 
mouth,  and  the  act ;  three  principal  things  in  Paradise, — glory, 
riches,  and  justice ;  three  things  which  are  especially  displeas- 
ing to  God, — an  avaricious  rich  man,  a  proud  poor  man,  and  a 


NUMERICAL   IDEAS   OF   THE    ANCIENTS.  89 

luxurious  old  man ;  three  things  which  are  in  no  esteem, — the 
strength  of  a  porter,  the  advice  of  a  poor  man,  and  the 
beauty  of  a  beautiful  woman.  And  all  things,  in  short,  are 
founded  in  three,  that  is,  in  number,  in  weight,  and  in  meas- 
ure." 

In  these  fanciful  speculations,  the  number  seven  has  received 
an  equal,  if  not  a  greater  distinction  than  the  number  three. 
In  the  year  1502,  there  was  printed  at  Leipsic  a  work  in  honor 
of  the  number  seven,  especially  composed  for  the  use  of  the 
students  of  the  university,  which  consisted  of  seven  parts, 
each  part  consisting  of  seven  divisions.  In  1624,  William 
Ingpen,  Gent.,  of  London,  published  a  work  entitled  "  The 
Secrets  of  Numbers,  according  to  Theological,  Arithmetical, 
Geometrical,  and  Harmonical  Computation.  Drawn  for  the 
better  part,  out  of  those  ancients,  as  well  as  Neoteriques. 
Pleasing  to  read,  profitable  to  understand,  opening  themselves 
to  the  capacities  of  both  learned  and  unlearned,  being  no  other 
than  a  key  to  lead  men  to  any  doctrinal  knowledge  whatso- 
ever." Di  Borgo  seems  to  have  been  influenced  by  the  same 
principle  in  determining  the  number  of  the  divisions  of  arith- 
metic; for  he  says:  "The  ancient  philosophers  assign  nine  parts 
of  algorism,  but  we  will  reduce  them  to  seven,  in  reverence  of 
the  seven  gifts  of  the  Holy  Spirit;  namely,  numeration, 
addition,  subtraction,  multiplication,  division,  progressions, 
and  extraction  of  roots." 

Some  of  these  fancies  are  not  entirely  extinct  at  the  present 
day.  In  England,  seven  constitutes  the  term  of  apprenticeship, 
the  period  for  academical  degrees,  and  as  in  our  own  country, 
the  product  of  these  two  magic  numbers  three  and  seven  con- 
stitutes the  legal  age  of  majority ;  and  the  frequent  use  of  the 
number  seven  in  the  Bible  has  given  it  associations  which 
have  caused  it  to  be  regarded  as  a  sacred  number. 


SECTION   II. 

1RITHMETICAL  LANGUAGE 


I.  Numeration. 

II.  Notation. 

III.  Origin  of  Symbols. 

IY.  Basis  of  the  Scale. 

V.  Other  Scales  of  Numeration 

VI.  A  Duodecimal  Scale. 

VII.  Greek  Arithmetic. 

VIII.  Roman  Arithmetic. 

IX.  Palpable  Arithmetic. 


CHAPTER  I. 

NUMERATION,  OR   THE   NAMING   OF   NUMBERS. 

BEGINNING  at  the  Unit,  we  obtain,  by  a  process  of  syn- 
thesis, arithmetical  objects  which  we  call  Numbers. 
These  objects  we  distinguish  by  names,  and  thus  obtain  the 
language  of  arithmetic.  This  language  is  both  oral  and 
written.  The  oral  language  of  arithmetic  is  called  Numera- 
tion ;  the  written  language  of  arithmetic  is  called  Notation. 
Numeration  treats  of  the  method  of  naming  numbers;  Nota- 
tion treats  of  the  method  of  writing  numbers.  As  oral 
language  always  precedes  written  language,  it  is  seen  that 
Numeration  precedes  Notation,  and  that  the  practice  of  arith- 
meticians in  reversing  this  order  is  illogical. 

Numeration  is  the  method  of  naming  numbers.  It  also 
includes  the  reading  of  numbers  when  expressed  by  characters. 
The  oral  language  of  arithmetic  is  based  upon  a  principle 
peculiarly  simple  and  beautiful.  Instead  of  giving  independ- 
ent names  to  the  different  numbers,  which  would  require  more 
words  even  to  count  a  million  than  one  could  acquire  in  a  life- 
time, we  name  a  few  of  the  first  numbers,  and  then  form  groups 
or  collections,  name  these  groups  or  collections,  and  then  use 
the  first  simple  names  to  number  the  groups.  The  method  13 
really  that  of  classification,  which  performs  for  arithmetic 
somewhat  the  same  service  of  simplification  that  it  does  in 
natural  science.  This  ingenious,  though  simple  and  natural 
method  of  breaking  numbers  up  into  classes  or  groups,  seems 
to  have  been  adopted  by  all  nations.  With  the  civilized  world 
and  with  most  uncivilized  tribes,  these  groups  generally  con- 

(93) 


94  THE    PHILOSOPHY   OF   ARITHMETIC. 

sist  of  ten  single  things,  suggested,  undoubtedly,  by  the 
practice  among  primitive  races,  of  reckoning  by  counting  the 
fingers  of  the  two  hands. 

Method  of  Naming. — The  fundamental  principle  of  naming 
numbers,  then,  is  that  of  grouping  by  tens.  We  regard  ten 
single  things  as  forming  a  single  collection  or  group;  ten  of 
these  groups  forming  a  larger  group,  and  so  on;  ten  groups 
of  any  one  value  forming  a  new  group  of  ten  times  the  value, 
each  group  being  regarded  and  used  as  a  single  thing.  In  this 
way,  by  giving  names  to  the  first  nine  numbers,  and  names  to 
the  groups,  and  employing  the  first  nine  to  number  the  groups, 
we  are  enabled  to  express  the  largest  numbers  in  a  concise  and 
convenient  form.  The  value  of  this  method  of  naming  may 
be  seen  from  the  consideration  that,  without  it,  the  memory 
would  be  overwhelmed  by  the  multiplicity  of  disconnected 
words,  and  we  should  require  a  lifetime  to  learn  the  names  of 
numbers,  even  up  to  a  few  hundred  thousands.  It  also  enables 
us  to  form  a  clear  and  distinct  conception  of  large  numbers, 
whose  composition  we  discover  in  the  words  by  which  they 
are  expressed,  or  in  the  symbols  by  which  they  are  represented. 
It  serves,  also,  as  a  basis  for  the  ingenious  and  useful  method 
of  writing  numbers,  without  which  arithmetic  would  be  almost 
useless  to  us. 

Naming  numbers  in  this  way,  a  single  thing  is  called  one ; 
one  and  one  more  are  two  ;  two  and  one  more  are  three  ;  and 
in  the  same  manner  we  obtain  four,  five,  six,  seven,  eight,  and 
nine,  and  then  adding  one  more  and  collecting  them  into  a 
group,  we  have  ten.  Now,  regarding  the  collection  ten  as  a 
single  thing,  and  proceeding  according  to  the  principle  stated, 
we  have  one  and  ten,  two  and  ten,  three  and  ten,  etc.,  up  to 
ten  and  ten,  which  we  call  two  tens.  Continuing  in  the  same 
manner,  we  have  two  tens  and  one,  two  tens  and  two,  etc, 
up  to  three  tens,  and  so  on  until  we  obtain  ten  of  these  groups 
of  tens.  These  ten  groups  of  tens  we  now  bind  together  by  a 
thread  of  thought,  forming  a  new  group  which  we  call  a  hun~ 


NUMERATION,  OR   THE   NAMING   OF   NUMBERS.  95 

dred.  Proceeding  from  the  hundred  in  the  same  way,  we 
unite  ten  of  these  into  a  larger  group  which  we  name  thousand, 
etc. 

This  is  the  actual  method  by  which  numbers  were  originally 
named;  but  unfortunately,  perhaps,  for  the  learner  and  for  sci- 
ence, some  of  these  names  have  been  so  much  modified  and 
abbreviated  by  the  changes  incident  to  use,  that,  with  several 
of  the  smaller  numbers  at  least,  the  principle  has  been  so  far 
disguised  as  not  to  be  generally  perceived.  If,  however,  the 
ordinary  language  of  arithmetic  be  carefully  examined,  it  will 
be  seen  that  the  principle  has  been  preserved,  even  if  disguised 
so  as  not  always  to  be  immediately  apparent.  Instead  of  one 
and  ten  we  have  substituted  the  word  eleven,  derived  from  an 
expression  formerly  supposed  to  mean  one  left  after  ten,  but 
now  believed  to  be  a  contraction  of  the  Saxon  endlefen,  or 
Gothic  ainlif  (ain,  one,  and  lif,  ten);  and  instead  of  two  and 
ten,  we  use  the  expression  twelve,  formerly  supposed  to  have 
been  derived  from  an  expression  meaning  two  left  after  ten, 
but  now  regarded  as  arising  from  the  Saxon  twelif  or 
Gothic  tvalif  (tva,  two,  and  lif,  ten.) 

With  the  numbers  following  twelve,  the  principle  can  be 
more  readily  seen,  though  by  constant  use  the  original  expres- 
sions have  been  abbreviated  and  simplified.  The  stream  of 
speech,  "  running  day  by  day,"  has  worn  away  a  part  of  the 
primary  form,  and  left  us  the  words  as  they  now  exist.  Thus, 
supposing  the  original  expression  to  be  three  and  ten,  (orig- 
inally the  Anglo-Saxon  thri  and  tyn)  if  we  drop  the  conjunction 
and,  we  shall  have  three-ten;  changing  the  ten  to  teen  we 
have  three-teen;  then  changing  the  three  to  thir,  and  omitting 
the  hyphen,  we  have  the  present  form  thirteen.  In  a  similar 
manner  the  expression  four  and  ten  becomes  fourteen;  five 
and  ten,  fifteen  ;  six  and  ten,  sixteen,  etc.  By  the  same  prin- 
ciples of  abbreviation  and  euphonic  change,  we  might  have 
obtained  twenty,  thirty,  etc.  Supposing  the  original  form  to 
be  two  tens,  or  twain  tens  (in  the  Saxon  twentig,  from  twegen, 


96  THE   PHILOSOPHY   OF   ARITHMETIC. 

two,  and  tig,  ten),  then  changing  the  twain  to  twen,  and  the 
tens  to  ty,  we  shall  have  the  common  form,  twenty.  In  three 
tens,  changing  the  three  to  thir  and  the  tens  to  ty,  we  have 
thirty.  In  the  same  way  we  obtain  forty,  fifty,  sixty,  etc., 
and  from  these  by  omitting  the  and  in  the  expression  two  tens 
and  one,  two  tens  and  two,  etc.,  we  have  twenty-one,  twenty- 
two,  thirty-three,  forty-seven,  etc. 

To  illustrate  the  law  of  the  formation  of  these  names,  we 
have  used  the  present  English  forms  rather  than  those  in  which 
the  transformations  actually  occurred.  It  will  be  remembered 
that  these  names  were  derived  from  the  Anglo-Saxon,  and  the 
changes  which  we  have  illustrated  took  place  in  that  language 
before  the  names  were  adopted  in  the  English  tongue.  The 
word  thirteen  was  actually  derived  from  the  Anglo-Saxon 
threo-tyne,  which  was  composed  of  thri,  three,  and  tyne,  ten ; 
fourteen  from  feowertyne,  composed  of  feower,  four,  and  tyne, 
ten,  etc.  We  get  the  word  twenty  from  the  Anglo-Saxon 
twentig,  which  is  composed  of  the  Anglo-Saxon  twegen,  two, 
and  tig,  ten ;  thirty  from  thritig,  which  is  composed  of  thri, 
three,  and  tig,  ten,  etc.  The  law  of  the  composition  of  these 
original  words  is  no  doubt  the  same  as  that  illustrated  by 
the  use  of  the  English  words  given  above. 

In  a  similar  manner  we  name  the  numbers  from  one  hundred 
to  the  next  group,  consisting  of  ten  hundreds,  to  which  we 
assign  a  new  name,  calling  it  thousand.  After  reaching  the 
thousand,  a  change  occurs  in  the  method  of  grouping.  Previ- 
ously, ten  of  the  old  groups  made  one  of  the  next  higher  group, 
but  after  the  third  group,  or  thousands,  it  requires  a  thousand 
of  an  old  group  to  form  a  new  group,  which  receives  a 
new  name.  A  thousand  thousands  forms  the  next  group 
after  thousands,  which  we  call  million  from  the  Latin  mille, 
a  thousand.  In  the  same  manner,  one  thousand  millions 
gives  a  new  group  which  we  call  billion,  one  thousand  billions 
a  new  group  which  we  call  trillion,  etc. 

This  change  in  the  law  by  which  a  new  group  is  formed  from 


NUMERATION,  OE   THE   NAMING   OF   NUMBERS.  97 

an  old  one,  is  not  an  accident;  it  is  intentional.  It  is  due  to 
science,  rather  than  to  chance.  The  method  of  counting  ten 
in  a  group  was  commenced  in  an  age  anterior  to  science,  and 
Droceeded  no  farther  than  hundreds  and  thousands,  since  the 
wants  of  the  people  did  not  require  larger  numbers ;  but  when 
arithmetic  began  to  be  cultivated  as  a  science,  it  was  seen  to 
be  a  matter  of  convenience  to  increase  the  size  of  the  groups 
receiving  a  new  name,  and  then  the  law  became  changed. 

The  reason  that  the  law  of  naming  numbers  does  not  appear 
in  the  names  of  the  smaller  numbers,  is,  that  they  became 
changed  from  the  original  form  on  account  of  their  frequent 
use.  The  same  fact  appears  in  grammar  in  the  irregularity  of 
the  verbs  expressing  ordinary  actions,  as  run,  go,  eat,  drink, 
etc.,  which  became  thus  irregular  in  the  formation  of  their 
tenses  from  the  constant  and  careless  use  of  the  common  peo- 
ple, before  the  language  was  fixed  by  the  rules  of  science  or 
the  art  of  printing. 

Utility. — The  utility  of  the  method  of  naming  numbers  by 
collecting  them  into  groups  or  bunches,  is  generally  imperfectly 
appreciated.  The  method  which  naturally  would  be  first  sug- 
gested to  the  mind,  is  to  give  each  number  an  independent 
name,  just  as  we  distinguish  rivers,  cities,  states,  etc.  This 
would,  of  course,  require  a  vocabulary  of  names  as  extensive 
as  the  series  of  natural  numbers, — a  vocabulary  which,  even  for 
the  ordinary  purposes  of  life,  could  be  learned  only  by  years 
of  labor.  By  the  method  of  groups,  the  vocabulary  is  so  sim- 
ple that  it  can  be  acquired  and  employed  with  the  greatest 
ease.  It  may  be  remarked,  that  this  method  of  grouping, 
though  suggested  by  the  accidental  circumstance  of  counting 
the  fingers,  is  in  accordance  with  that  universal  operation  of 
the  mind  by  which  it  binds  up  its  knowledge  into  bunches  or 
packages.  It  is,  in  fact,  based  upon  the  principle  of  generaliza- 
tion and  classification. 

Origin  of  Names. — The  origin  or  primary  meaning  of  the 
names  applied  to  the  first  ten  numbers,  is  not  known.     It  has 
7 


98  THE    PHILOSOPHY   OF   ARITHMETIC. 

been  supposed  that  the  names  of  the  simple  numbers  were 
originally  derived  from  some  concrete  objects,  and  there  are  a 
few  facts  which  seem  to  indicate  the  correctness  of  this  suppo- 
sition. Thus,  the  Persian  name  for  five  is  pendje,  while 
pentcha  means  the  expanded  hand,  and  the  corresponding  terms 
in  the  Sanskrit  are  said  to  have  a  similar  meaning.  The  term 
liniq,  which  with  slight  modifications  is  used  for  five  through- 
out the  Indian  Archipelago,  means  hand  in  the  language  of 
the  Otaheite  and  other  islands.  Among  the  Jaloffs,  an  African 
tribe,  the  word  for  five,  juorum,  likewise  signifies  hand. 
Among  the  Greenlanders  the  term  fox  twenty  is  innuk,  or  mam; 
that  is,  after  completing  the  counting  of  fingers  and  toes,  they 
say  innuk  or  man ;  and  there  are  also  examples  of  the  identity 
of  the  term  for  man  and  twenty  among  some  of  the  tribes  of 
South  America. 

Among  the  Indians  of  Bogota,  New  Grenada,  the  term 
quicha,  meaning  a  foot,  is  used  to  number  the  second  decade, 
while  twenty  is  named  gueta,  which  signifies  a  house.  Nearly 
all  the  South  American  tribes  use  the  word  for  hand  to  express 
five,  and  in  many  cases  the  word  for  man  is  used  to  express 
twenty.  A  tribe  in  Paraguay  denote  four  by  an  expression 
which  means  the  fingers  of  the  Emu,  a  bird  common  in  Par- 
aguay, possessing  four  claws  on  each  foot,  three  before,  and 
one  turned  back ;  and  their  word  for  five  is  the  name  of  a 
beautiful  skin  with  five  different  colors.  The  same  number  is, 
however,  more  commonly  expressed  by  hanam  begem,  the 
fingers  of  one  hand;  ten  is  expressed  by  the  fingers  of  both 
hands;  and  for  twenty  they  say  hanam  rihegem  cat  gracha- 
haka  anomichera  hegem,  the  fingers  of  both  hands  and  feet. 
Among  the  Caribbeans,  the  fingers  are  termed  the  children  of 
the  hand,  and  the  toes  children  of  the  feet ;  and  the  phrase  for 
ten,  chou  oucabo  raim,  means  all  the  children  of  the  hands 

Humboldt  has  given  from  the  researches  of  Duquesne,  the 
etvmological  signification  of  some  of  the  numerals  of  the 
Indians  of  New  Grenada.     Thus,  ata,  one,  signifies  water; 


NUMERATION,  OR    THE    NAMING   OF   NUMBERS.  99 

bosa,  two,  an  enclosure;  mica,  three,  changeable ;  muyhica% 
four,  a  cloud  threatening  a  tempest ;  hisa,  five,  repose ;  ta, 
six,  harvest;  cahupqua,  seven,  deaf;  suhuzza,  eight,  a  tail; 
and  ubchica,  ten,  resplendent  moon.  No  meaning  has  been 
discovered  for  aca,  the  numeral  for  nine.  It  would  seem  im- 
possible, amidst  such  various  meanings,  to  discover  any  prin- 
ciple which  may  seem  to  have  pointed  out  the  use  of  these 
terms  as  numerals. 

In  the  Mexican  numeral  symbols  there  is  an  intelligible  con- 
nection between  the  sign  and  the  thing  signified,  though  the 
association  seems  to  be  entirely  arbitrary.  Thus,  the  symbol 
for  one  is  a  frog;  for  two,  a  nose  with  extended  nostrils,  part 
of  the  lunar  disk,  figured  as  a  face ;  for  three,  two  eyes  open, 
another  part  of  the  lunar  disk ;  for  four,  two  eyes  closed  ; 
for  Jive,  two  figures  united,  the  nuptials  of  the  sun  and 
moon,  conjunction ;  for  six,  a  stake  with  a  cord,  alluding 
to  the  sacrifice  of  Guesa  tied  to  a  pillar ;  for  seven,  two  ears ; 
for  eight,  no  meaning  assigned ;  for  nine,  two  frogs  coupled ; 
for  ten,  an  ear ;  for  twenty,  a  frog  extended. 

The  following  theory,  advanced  by  Prof.  Goldstiicker,  in  a 
paper  read  before  the  Philological  Society  in  1870,  in  which  he 
gives  good  linguistic  evidence  in  support  of  the  origin  of  the 
Sanskrit  numerals,  and  consequently  of  our  own,  is  at  least 
plausible,  and  will  be  interesting:  One,  he  says,  is  "he,"  the 
third  personal  pronoun ;  two,  "diversity;"  three,  "that  which 
goes  beyond ;"  four,  "and  three,"  that  is,  "one  and  three;" 
five,  "coming  after;"  six,  "four,"  that  is,  "and  four,"  or  "two 
and  four;"  seven,  "following;"  eight,  "two  fours,"  or  "twice 
four;"  nine,  "that  which  comes  after"  (ch.  nava,  new);  ten, 
"two  and  eight."  Thus,  only  one  and  two  have  distinct  orig- 
inal meanings.  After  giving  these,  our  ancestors'  powers 
needed  a  rest;  then  they  made  three,  and  added  to  it  one  for 
four ;  then  took  another  rest,  repeated  the  notion  of  three  in 
five,  and  the  notion  of  four  in  six ;  then  rested  once  more, 
and  again  repeated  the  notion  of  three  and  five  in  seven  ;  took 


100  THE   PHILOSOPHY    OF   ARITHMETIC. 

another  rest,  and  got  a  new  idea  of  two  fours  for  eight;  but 
for  nine  repeated  for  the  fourth  time  the  "  coming  after"  notion 
of  three,  five,  and  seven  ;  while  for  ten  they  repeated  for  the 
third  time  the  addition-notion  of  four  and  six.  The  Professor 
insists  strongly  on  this  seeming  poverty  and  helplessness  of 
the  early  Indo-European  mind.  He  does  not  put  forward  the 
above  meanings  of  the  numerals  as  new,  though  he  believes 
that  his  history  of  most  of  the  forms  of  their  names  is  so. 
The  anomalous  form  of  the  Sanskrit  shash,  six — the  hardest  of 
them — first  set  him  at  work  on  the  numerals,  and  the  Zend 
form  kshvas  led  him  to  the  true  explanation  of  this,  and  thence 
to  that  of  the  other  numerals. 

In  closing  this  chapter,  we  remark  that  the  names  of  the 
periods  above  duodecillions  have  not  been  fully  settled  by  usage. 
Prof.  Henkle,  who  has  examined  the  subject  with  considerable 
care,  finds  a  law  which  he  maintains  should  hold  in  the  forma- 
tion of  the  names  of  the  higher  periods.  The  terms  quintillions, 
sextillions,  and  nonillions  are  formed,  not  from  the  cardinals, 
quinque,  sex,  and  novem,  but  from  the  ordinals,  quintus,  sextus, 
and  nonus.  From  this  he  infers  that  analogy  plainly  demands 
that  the  names  beyond  duodecillions  should  be  formed  from 
the  Latin  ordinal  numerals.  For  the  names  thus  formed,  see 
appendix. 


CHAPTER  II. 

NOTATION,  OR   THE   WRITING   OF   NUMBERS. 

ARITHMETICAL  language  is  the  expression  of  arithmet- 
ical ideas.  These  ideas  may  be  expressed  in  sound  to 
the  ear,  or  in  visible  form  to  the  eye ;  arithmetical  language  is, 
therefore,  both  oral  and  written.  The  oral  language  is  called 
Numeration ;  the  written  language,  Notation.  Numeration  is 
the  method  of  naming  numbers;  Notation  is  the  method  of 
writing  numbers.  From  this  consideration  it  would  seem  that 
the  written  language  of  arithmetic  must  bear  an  intimate  rela- 
tion to  the  oral  language,  which  we  find  to  be  the  case.  The 
general  method  of  writing  numbers,  now  adopted  by  all  civil- 
ized nations,  is  the  Hindoo,  usually  called  the  Arabic  method. 
This  method  is  based  upon,  and  arises  naturally  out  of,  the 
method  of  naming  numbers  by  groups. 

The  fundamental  principle  of  the  Arabic  system  is  the 
ingenious  and  refined  idea  of  place  value.  Recognizing  the 
method  of  naming  numbers  by  groups,  it  assumes  to  represent 
these  groups  by  the  simple  device  of  place.  It  fixes  upon  a 
few  characters  to  represent  a  few  of  the  first  numbers,  and 
then  employs  these  same  characters  to  number  the  groups, 
the  group  numbered  being  indicated  by  the  place  of  the  char- 
acter. This  leads  to  the  distinction  of  the  intrinsic  and  local 
value  of  the  numerical  characters.  Each  character  has  a  defi- 
nite value  when  it  stands  alone,  and  a  relative  value  when  used 
in  connection  with  other  characters. 

The  number  of  the  arithmetical  characters  is  determined  by 
the  number  of  units  in  the  group.     The  grouping  being   by 

(101) 


102  THE    PHILOSOPHY    OF    ARITHMETIC. 

tens,  the  number  of  characters  needed  is  only  nine,  one  less 
than  the  number  of  units  in  the  group.  These  characters  are 
called  digits,  from  the  Latin  digitus,  a  finger,  the  name  com- 
memorating the  ancient  custom  of  reckoning  by  counting  the 
fingers.  In  the  combination  of  these  characters  to  express 
numbers,  it  will  often  be  required  to  indicate  the  absence  of 
some  group ;  hence  arises  the  necessity  of  a  character  which 
expresses  no  value,  a  character  which  denotes  merely  the 
absence  of  value.  This  character  is  known  as  naught,  or  zero. 
We  thus  have  the  following  ten  characters :  1,  2,  3,  4,  5,  6,  7, 
8,  9,  0,  with  which  we  are  able  to  express  all  possible  numbers. 

Utility. — The  Arabic  system,  based  upon  the  refined  idea  of 
place  value,  is  one  of  the  happiest  results  of  human  intelli- 
gence, and  deserves  our  highest  admiration.  Remarkable  as  is 
its  simplicity,  it  constitutes,  regarded  in  its  philosophical  char- 
acter or  its  practical  value,  one  of  the  greatest  achievements  of 
the  human  mind.  In  the  hands  of  a  skillful  analyst,  it  be- 
comes a  most  powerful  instrument  in  wresting  from  nature  her 
hidden  truths  and  occult  laws.  Without  it,  many  of  the  arts 
would  never  have  been  dreamed  of,  and  astronomy  would  have 
been  still  in  its  cradle.  With  it,  man  becomes  armed  with 
prophetic  power, — predicting  eclipses,  pointing  out  new  planets 
which  the  eye  of  the  telescope  had  not  seen,  assigning  orbits 
to  the  erratic  wanderers  of  space,  and  even  estimating  the  ages 
that  have  passed  since  the  universe  thrilled  with  the  sublime 
utterance,  "Let  there  be  light!"  Familiarity  with  it  from  child- 
hood detracts  from  our  appreciation  of  its  philosophical  beauty 
and  its  great  practical  importance.  Deprived  of  it  for  a  short 
time,  and  compelled  to  work  with  the  inconvenient  methods  of 
other  systems,  we  should  be  able  to  form  a  truer  idea  of  the 
advantages  which  this  invention  has  conferred  on  mankind. 

Relation  to  Numeration. — Though  the  methods  of  notation 
and  numeration  are  intimately  related,  there  is  also  an  essential 
distinction  between  them.  Though  similar,  they  are  by  no  means 
identical  in  principle.       Their  similarity  is  seen  in  the  fact  that 


NOTATION,  OR   THE   WRITING    OF   NUMBERS.  103 

the  method  of  notation  could  not  be  applied  without  the  method 
of  numbering  by  groups;  their  distinction  is  seen  in  the  fact 
that  we  could  have  the  present  method  of  numeration  without 
the  Arabic  system  of  notation.  The  notation  seems  to  be  an 
immediate  outgrowth  from  the  numeration,  yet  not  a  necessary 
one;  for  many  nations  who  had  the  same  method  of  naming 
numbers,  employed  other  methods  of  writing  them. 

Their  true  relation  also  appears  in  considering  their  common 
relation  to  the  decimal  scale.  The  decimal  principle  belongs 
both  to  our  method  of  naming  and  of  writing  numbers.  This 
coincidence  is  not  accidental,  but  essential  to  the  harmony 
of  oral  and  written  expression.  The  necessity  of  this  would 
be  very  apparent  if  we  should  attempt  to  change  the  base 
of  the  scale  of  notation  without  changing  the  base  of  the 
method  of  naming  numbers.  With  our  present  base  we  say 
one  and  ten,  two  and  ten,  etc.,  or  at  least  their  equivalents; 
and  our  written  expressions  are  read  in  the  same  manner. 
Should  we  adopt  any  other  scale  of  notation,  retaining  our 
present  base  in  naming  numbers,  the  reading  of  numbers  in 
this  new  scale  would  be  so  awkward  and  inconvenient  as  to  be 
almost  impossible.  Hence  it  follows,  that  for  a  scale  of  nota- 
tion to  be  advantageously  employed,  the  methods  of  naming 
and  writing  numbers  should  possess  the  same  basis.  Thus,  if 
the  scale  of  notation  be  quinary,  instead  of  naming  numbers 
Jive,  six,  seven,  etc.,  we  should  say  five,  one  and  five,  two  and 
five,  etc.;  if  the  scale  were  senary,  we  should  say  six,  one  and 
six,  two  and  six,  etc. 

Relation  to  the  Base. — It  will  also  be  seen  that  the  princi- 
ple of  the  methods  of  naming  and  writing  numbers  is  entirely 
distinct  from  the  number  used  as  the  base.  The  intimate  asso- 
ciation of  the  Arabic  system  with  the  base,  has  sometimes  led 
to  the  idea  that  the  base  is  a  part  of  the  system  itself.  This 
error  should  be  carefully  avoided.  The  Arabic  method 
assumes  that  we  name  numbers  by  groups,  and  that  each 
group  contains  ten;  but  it  is  in  principle  entirely  independent 


104  THE    PHILOSOPHY    OF    ARITHMETIC. 

of  the  number  constituting  a  group.  The  number  in  the  group 
determines  the  base  of  the  scale,  and  consequently  the  number 
of  characters  to  be  used,  but  does  not  affect  the  principle  of 
the  method,  which  is  simply  that  of  place  value.  Should  we 
change  the  base  of  numbering,  it  would  change  the  names  of 
the  numbers  after  twelve,  and  the  base  of  the  Arabic  scale ; 
but  it  would  in  no  wise  affect  the  principle  of  either  the  method 
of  numeration  or  of  notation. 

Number  of  Characters. — The  number  of  characters  in  the 
Arabic  system  of  notation  depends  upon  the  number  of 
units  in  the  groups  of  numeration.  Thus,  we  must  have  as 
many  simple  characters  as  will  express  the  different  numbers 
from  one  until  we  reach  within  a  unit  of  the  group.  We  shall 
have  no  character  for  the  group,  since,  according  to  the  device 
of  place  value,  it  is  to  be  indicated  by  changing  the  place  of 
the  symbol  which  represents  one,  it  being  one  of  the  first 
group.  The  number  of  significant  characters  must,  therefore, 
be  always  one  less  than  the  number  denoting  the  base  of  the 
system.  In  the  decimal  scale  the  number  of  digits  is  nine ; 
in  an  octary  scale  it  would  be  seven ;  in  a  quinary  scale,  four, 
etc. 

Origin. — The  origin  of  this  system  of  notation  is  now  uni- 
versally accredited  to  the  Hindoos.  When,  by  whom,  and  how 
it  was  invented,  we  do  not  know.  It  is  not  improbable  that  it 
began  with  the  representation  of  the  spoken  words  by  marks, 
or  abstract  characters.  They  may  at  first  have  given  inde- 
pendent characters  to  the  numbers  as  far  as  represented.  It 
then  probably  occurred  to  them  that,  since  they  gave  independ- 
ent names  to  a  few  numbers  and  then  numbered  by  groups, 
they  could  simplify  their  system  of  notation  by  making  it  cor- 
respond to  their  system  of  numeration.  Then  first  dawned 
upon  the  mind  the  idea  of  a  few  characters  to  represent  the 
first  simple  numbers,  and  the  use  of  these  same  characters  to 
number  the  groups.  They  now  stood  on  the  threshold  of  one 
of  the  greatest  discoveries  of  all  time.     Here  arose  the  ques- 


NOTATION,  OR    THE    WRITING   OF   NUMBERS.  105 

tion— How  are  these  groups  to  be  distinguished?  How  shall 
we  determine  when  a  character  denotes  a  number  of  units  or 
tens,  or  hundreds,  etc.?  How  many  methods  occurred  to  them 
before  the  method  of  place,  who  can  tell  ?  This  might  have 
been  done  by  slightly  varying  the  character,  by  attaching  some 
mark  to  it,  by  annexing  the  initial  of  the  group,  etc.;  either  of 
which  would  have  been  comparatively  complicated  and  incon- 
venient. At  last,  to  the  mind  of  some  great  thinker,  occurred 
the  simple  idea  of  place  value,  and  the  problem  was  solved. 
"Who  was  the  man  ?"  is  a  question  answered  only  by  its  own 
echo,  for  his  name  sleeps  in  the  silence  of  the  past.  Were  it 
known,  mankind  would  feel  like  rearing  a  monument  to  his 
memory,  as  high  and  enduring  as  the  Pyramids  of  Egypt;  but 
now  it  can  only  raise  its  altar  to  the  Unknown  Genius. 

Origin  of  Characters. — The  origin  of  the  characters,  like 
that  of  the  system,  is  shrouded  in  mystery  ;  but  little  light 
upon  the  subject  comes  down  the  historic  path.  Many  of  the 
early  writers  gave  some  ingenious  speculations  concerning 
their  origin.  Gatterer  imagined  that  he  had  discovered  in 
Egyptian  manuscripts  written  in  the  enchoriac  character, 
that  the  digits  were  denoted  by  nine  letters;  and  Wachter 
supposed  them  to  have  a  natural  origin  in  the  different  com- 
binations of  the  fingers:  thus,  unity  is  expressed  by  the 
outstretched  finger  ;  two  by  two  fingers,  which  may  have  been 
represented  by  two  marks  that,  by  long  use,  passed  into  the 
present  form,  and  so  on  for  all  the  other  symbols.  In  the 
absence  of  facts,  three  theories  have  been  presented,  which  are 
at  least  interesting  on  account  of  their  ingenuity,  and  are 
certainly  somewhat  plausible.  One  of  these  theories  is  that 
they  are  formed  by  the  combination  of  straight  lines,  as  the 
primary  representation  of  numbers ;  another  is  that  they  are 
formed  by  the  combination  and  modification  of  angles;  and 
still  another  and  more  recent  theory  is  that  they  are  the 
initial  letters  of  the  Hindoo  numerals.  These  three  theories 
may  be  distinguished  as  the  theories  of  lines,  angles,  and 
initial  letters. 


106  THE    PHILOSOPHY    OF    ARITHMETIC. 

The  first  theory  is  based  on  the  primary  use  of  straight 
lines  to  represent  numbers.  By  this  method,  one  straight 
line,  |,  would  represent  one;  two  straight  lines  which  may 
have  been  connected  thus,  L,  two ;  three  lines,  thus,  ^,  or  with 
a  connecting  curve,  thus,  3,  three;  four  lines  arranged  thus, 
Q  or  thus, 2],  four ;  five  lines  arranged  thus,  £3,  five;  six  lines 
arranged  thus,  [3,  six;  seven  lines,  thus,  [J  seven  ;  eight  lines 
thus,  9,  or  thus  Y,  eight;  nine  lines,  thus,  ^»  nine.  The 
zero  is  supposed  to  have  been  originally  a  circle,  suggested 
from  counting  around  the  fingers  and  thumbs  held  in  a  circular 
position. 

The  second  theory  is  based  upon  the  use  of  angles  to  repre- 
sent numbers.  The  ancient  mathematicians  were  noted  for 
their  astronomical  observations  and  calculations,  and  being 
thus  familiar  with  the  use  of  angles,  it  is  not  unreasonable  to 
suppose  that  they  would  employ  the  angle  in  their  representa- 
tion of  numbers.  Thus,  they  might  very  naturally  have  used 
one  angle,  'J,  for  one ;  two  angles,  2»  f°r  iw0  >'  three  angles  ]£, 
for  three;  four  angles,  A  for  four;  five  angles,  g,  for  five, 
six  angles,  £,  for  six;  seven  angles,  9,  for  seven;  eight 
angles,  0,  for  eight ;  nine  angles,  ^,  for  nine.  These  char- 
acters being  frequently  made,  would  eventually  assume  the 
rounded  form  which  they  now  possess.  By  this  theory,  the 
character  for  zero  is  easily  and  naturally  accounted  for  If 
angles  were  used  to  represent  numbers,  nothing  would  be  rep- 
resented by  a  character  having  no  angles,  which  is  the  closed 
curve. 

The  latest  and  most  plausible  theory  for  the  origin  of  Arabic 
characters  is,  that  they  were  originally  the  initial  letters  of 
the  Sanskrit  numerals.  This  theory  is  presented  by  Prin- 
seps,  a  profound  Sanskrit  scholar,  and  is  indorsed  by  Max 
Miiller.  Such  a  use  of  initial  letters  was  entirely  feasible 
in  the  Sanskrit  language,  as  each  numeral  began  with  a  dif- 
ferent letter.  The  plausibility  of  the  theory  further  appears 
from  the  fact  that  it  follows  the  general  law  of  representing 


NOTATION,  OR   THE   WRITING   OF   NUMBERS.  107 

numbers  by  letters,  as  in  the  Roman,  Greek,  and  Hebrew 
systems. 

This  theory  does  not  account  for  the  origin  of  the  zero,  the 
most  important  character  of  them  all, — in  fact,  the  key  to  the 
system  of  modern  arithmetic.  No  other  system  of  notation 
except  the  sexagesimal  system,  had  it.  Max  Miiller  says : 
"  It  would  be  highly  important  to  find  out  at  what  time  the 
naught  first  occurs  in  Indian  inscriptions.  That  inscription 
would  deserve  to  be  preserved  among  the  most  valuable  monu- 
ments of  antiquity,  for  from  it  would  date  in  reality  the 
beginning  of  true  mathematical  science — impossible  without 
the  naught — nay,  the  beginning  of  all  the  exact  sciences  to 
which  we  owe  the  invention  of  telescopes,  steam  engines,  and 
electric  telegraphs."  Dr.  Peacock  supposes  that  it  was  derived 
from  the  Greek  o,  introduced  by  Ptolemy  to  denote  the  vacant 
places  in  the  sexagesimal  arithmetic;  the  Hindoos,  he  says, 
having  used  a  dot  for  this  purpose. 

It  seems  to  have  been  difficult  at  first  to  comprehend  the  pre- 
cise force  of  the  cipher,  which,  insignificant  in  itself,  serves  only 
to  determine  the  rank  and  value  of  the  other  figures.  When 
they  were  first  introduced  into  Europe,  it  was  deemed  necessary 
to  prefix  to  any  work  in  which  they  were  used,  a  short  treatise 
on  their  nature  and  application.  These  notices  are  often  met 
with  attached  to  old  vellum  almanacs,  or  inserted  in  the  blank 
leaves  of  missals,  and  frequently  intermixed  with  famous  prophe- 
cies, most  direful  prodigies,  and  infallible  remedies  for  scalds 
and  burns.  A  sort  of  mystery,  which  has  imprinted  its  trace 
on  our  language,  seemed  to  hang  over  the  practice  of  using 
the  cipher;  and  we  still  speak  of  deciphering  and  writing  in 
cipher,  in  allusion  to  some  dark  or  concealed  art.  Indeed,  in 
the  early  history  of  arithmetic  in  Europe,  either  on  account  of 
its  association  with  the  infidel  Mohammedans  from  whom  it 
was  derived,  or  of  the  popular  prejudice  against  learning  which 
prevailed  at  that  time,  the  system  was  regarded  as  belonging 
to  black  art  and  the  devil ;  and  it  was,  no  doubt,  this  popular 
prejudice  that  delayed  its  general  introduction  into  Christian 
Europe. 


CHAPTER  III. 

ORIGIN   OP   ARITHMETICAL    SYMBOLS. 

THE  symbols  of  arithmetic  may  be  divided  into  three  general 
classes:  Symbols  of  Number,  Symbols  of  Operation,  and 
Symbols  of  Relation.  What  is  the  origin  of  these  symbols  ; 
who  invented  them,  or  first  employed  them?  This  question,  a 
very  interesting  one,  I  shall  endeavor  to  answer  in  the  present 
chapter. 

I.  Symbols  of  Number. — The  Symbols  of  Number  em- 
ployed by  different  nations,  are  the  Arabic  figures  and  the 
letters  of  the  alphabet.  Nearly  all  civilized  nations  seem  to 
have  made  use  of  the  letters  of  the  alphabet  to  represent  num- 
bers. The  Greeks  divided  their  letters  into  several  classes,  to 
represent  the  different  groups  of  the  arithmetical  sctJe.  The 
Roman  system  employed  the  seven  letters,  I,  V,  X,  L,  C,  D, 
and  M,  to  represent  numbers.  The  Arabs  at  first  used  the 
Greek  method,  and  afterward  exchanged  it  for  that  of  the  Hin- 
doos. 

There  are  three  theories  given  for  the  origin  of  the  Arabic 
symbols  of  notation,  known  respectively  as  the  theory  of  lines, 
of  angles,  and  of  initial  letters.  These  three  theories  are 
explained  in  the  chapter  on  Notation.  It  may  also  be  remarked 
that  some  of  the  Arabian  authors  who  treat  of  astrological 
signs,  allege  that  the  Indian  or  Arabic  numerals  were  derived 
from  the  quartering  of  the  circle,  and  Leslie  says  that  the 
resemblance  of  these  natural  marks  to  the  derivative  ones 
appears  very  striking.  The  Roman  symbols  are  supposed  to 
have  originated  in  the  use  of  simple  straight  lines  or  strokes, 

(108) 


ORIGIN    OF    ARITHMETICAL   SYMBOLS.  109 

variously  combined,  for  which  were  subsequently  substituted  the 
letters  of  the  alphabet.  This  theory  is  explained  at  length  in  the 
chapter  on  Roman  Notation. 

Symbols  of  Operation. — The  Symbols  of  Operation  are  the 
signs  of  addition,  subtraction,  multiplication,  division,  involution, 
evolution,  and  aggregation.  The  origin  of  most  of  these  symbols 
has  been  definitely  determined. 

The  Symbols  of  Addition  and  Subtraction  were  first  introduced 
as  symbols  of  operation  by  Michael  Stifel,  a  German  mathematic- 
ian, in  a  work  entitled  Arithmetica  Integra,  published  at  Nurem- 
burg  in  1544.  These  signs  had  appeared  previously  in  a  work  of 
Johann  Widmann  called  the  Mercantile  Arithmetic,  published  at 
Leipzig  in  1489.  They  are,  however,  not  used  by  him  as  symbols 
of  operation,  but  merely  as  marks  of  excess  or  deficiency.  The 
next  oldest  book  extant  in  which  these  signs  are  found  is  that  of 
Christopher  Rudolff,  published  in  1524,  though  he  does  not  use 
them  as  symbols  of  operation.  Stifel  was  a  pupil  of  Rudolff,  and 
it  is  supposed  that  he  obtained  the  symbols  from  him,  for,  as  he 
himself  admits,  he  took  a  large  part  of  his  work  from  that  of 
Rudolff.  Stifel  introduces  the  symbols  as  if  he  had  originated 
them  or  their  new  use,  for  he  says,  "  thus,  we  place  this  sign," 
etc.,  and  "  we  say  that  the  addition  is  thus  completed,"  etc.  To 
Stifel,  therefore,  belongs  the  credit  of  first  using  these  symbols  as 
signs  of  operation. 

Why  these  particular  signs  were  adopted  has  been  a  matter  of 
conjecture.  Prof.  Rigaud  supposed  that  +  was  a  corruption  of 
P,  the  initial  for  plus,  and  Dr.  Davis  thought  that  it  was  a  cor- 
ruption of  et  or  fy.  Stifel,  however,  does  not  call  the  signs  plus 
and  minus,  but  signum  additorum  and  signum  subtractorum,  which 
renders  these  suppositions  improbable.  Dr.  Ritchie  suggested 
that  perhaps  +  was  two  marks  joined  together  in  addition,  and 
that  —  was  taken  to  indicate  subtraction,  since  it  is  ivhat  is  left 
after  one  of  the  marks  is  removed.  De  Morgan  thought  that  the 
minus  sign  —  was  first  used,  and  that  +  was  derived  from  it  by 
putting  a  small   cross-bar  for   distinction.     "  The    sign  +,"   he 


110  THE   PHILOSOPHY   OF   ARITHMETIC. 

says,  "  in  the  hands  of  Stifel's  printer  has  the  vertical  bar  much 
shorter  than  the  other,  and  when  it  is  introduced  into  the  wood- 
cuts of  the  engraver,  the  disproportion  is  greater  still."  The 
Hindoos,  from  whom  our  knowledge  of  algebra  was  originally 
derived,  used  a  dot  for  subtraction,  and  the  absence  of  the  dot  for 
addition,  and  De  Morgan  suggests  that  the  Hindoo  dot  may  have 
been  elongated  into  a  bar  to  signify  subtraction,  and  that  an  addi- 
tional line  to  the  symbol  of  subtraction  gave  the  sign  of  addition. 
M.  Libri  attributes  the  invention  of  -f  and  —  to  Leonardo  da 
Vinci,  the  celebrated  Italian  artist  and  philosopher,  but  it  is 
probable  that  Da  Vinci  used  the  symbol  +  for  the  figure  4. 

The  most  recent  explanation  of  these  signs  is  that  they  were 
originally  warehouse  marks.  In  Widmann's  arithmetic  they  occur 
almost  exclusively  in  practical  mercantile  questions.  Goods  were 
sold  in  chests  which  when  full  were  expected  to  hold  a  certain 
established  weight.  Any  excess  or  deficiency  was  indicated  by 
-f  or  — ,  and  these  signs  may  have  been  marked  with  chalk  on 
the  chests  as  they  came  from  the  warehouses.  Usually  the 
weight  of  the  chest,  it  may  be  supposed,  would  be  deficient,  and 
this  was  marked  with  the  sign  — ;  when  a  cask  or  chest  was 
above  the  standard  weight  the  line  —  may  have  been  crossed  with 
a  vertical  line  giving  the  symbol  +.  It  may  be  remarked  that 
these  symbols  were  not  immediately  adopted  by  mathematicians. 
In  a  work  on  algebra,  published  in  1619,  the  signs  of  addition  and 
subtraction  are  P  and  M  with  strokes  drawn  through  them. 

The  Symbol  of  Multiplication  (X),  St.  Andrew's  cross,  was  in- 
troduced by  William  Oughtred,  an  eminent  English  mathematic- 
ian and  divine,  born  at  Eton  in  1573.  The  work  in  which  this 
symbol  first  appeared  was  entitled  Clavis  Mathematical  "  Key  of 
Mathematics,"  and  published  in  1631.  Oughtred  was  a  fine 
thinker,  and  was  honored  by  the  title  "  prince  of  mathematicians." 
What  led  to  this  particular  form  for  the  symbol  is  unknown.  Two 
other  signs  for  multiplication  were  proposed, — the  (.)  by  Descartes 
and  the  curve  (^)  by  Leibnitz,  but  though  having  the  authority 
of  great  names  they  failed  of  adoption. 


ORIGIN    OF    ARITHMETICAL   SYMBOLS.  Ill 

The  Symbol  of  Division  (-*-)  was  introduced  in  1630  by  Dr. 
John  Pell,  Professor  of  Philosophy  and  Mathematics  at  Breda. 
This  symbol  was  used  by  some  old  English  writers  to  denote  the 
ratio  or  relation  of  quantities.  I  have  also  noticed  it  used  thus  in 
some  old  German  mathematical  works.  The  Arabs  used  a  dash, 
writing  one  number  under  the  other,  in  the  form  of  a  fraction. 
Dr.  Pell  was  highly  regarded  as  a  mathematician.  It  was  to  him 
that  Newton  first  explained  his  invention  of  fluxions. 

The  System  of  Exponents,  to  represent  the  powers  of  a  num- 
ber, has  been  generally  ascribed  to  Descartes,  1596-1650,  the 
illustrious  metaphysician  and  inventor  of  Analytical  Geometry. 
Exponents  were,  however,  employed  by  De  la  Roche  as  early  as 
1520,  but  Descartes'  extensive  use  of  them  led  to  their  general 
adoption.  The  earliest  writer  on  algebra  denoted  the  powers  of  a 
number  by  an  abbreviation  of  the  name  of  the  power.  Harriot,  a 
mathematician  of  the  17th  century,  repeated  the  quantity  to  indi- 
cate the  power  ;  thus,  for  a*  he  wrote  aaaa. 

The  Radical  Sign  (/)  was  introduced  by  Stifel,  the  introducer 
of  +  and  —  This  symbol  is  a  modification  of  the  letter  r,  the 
initial  of  radix,  root.  The  root  of  a  quantity  was  formerly  de- 
noted by  writing  the  letter  r  before  it,  and  this  letter  was  grad- 
ually changed  to  the  form  V • 

The  Vinculum  or  Bar,  placed  over  quantities  to  connect  them 
together,  thus,  4X3  +  5,  was  first  used  by  Vieta  in  1591,  the  in- 
troducer, in  algebra,  of  the  system  of  representing  known  quanti- 
ties by  symbols.  The  Parenthesis  and  Brackets  were  first  used  by 
Albert  Girard,  a  Dutch  writer  on  algebra,  in  1629. 

III.  Symbols   of  Relation Symbols  of  Relation  are  the 

signs  of  equality,  ratio,  equal  ratios,  inequality  and  deduction. 
The  origin  of  a  few  of  these  has  been  ascertained. 

The  Symbol  of  Equality  (=)  was  introduced  by  Robert  Re- 
corde,  an  English  physician  and  mathematician  of  the  sixteenth 
century.  It  first  appeared  in  1556  in  his  work  on  algebra,  called 
by  the  odd  title  The  Whetstone  of  Witte. 

This  sign  was  also  employed  by  Albert  Girard.     The  French 


112  THE   PHILOSOPHY   OF   ARITHMETIC. 

and  German  mathematicians  used  the  symbol  oo  to  denote  equality, 
even  long  after  Recorde.  This  symbol  is  said  to  be  a  modifica- 
tion of  the  diphthong  «,  the  initial  of  the  Latin  phrase  cequale  est. 
It  is  also  stated  that  the  symbol  =  was  often  used  as  an  abbrevia- 
tion for  est  in  mediaeval  manuscripts. 

The  Symbol  of  Ratio  (:)  is  supposed  to  be  a  modification  of  the 
sign  of  division.  The  sign  of  division  was  frequently  employed 
by  the  old  English  and  German  mathematicians  to  indicate  the 
relation  of  quantities.  Who  first  omitted  the  dash  and  employed 
the  present  form  of  the  symbol  of  ratio,  I  have  not  been  able  to 
ascertain.     It  occurs  in  a  work  by  Clairaut,  published  in  1760. 

The  Symbol  of  Equal  Ratios  (:  :)  may  be  a  modification  of  the 
sign  of  equality  (=)  or  a  duplication  of  the  symbol  of  ratio  (:), 
but  this  is  not  certain.  It  seems  to  have  been  introduced  by 
Oughtred,  in  a  work  published  in  1631,  and  was  brought  into 
common  use  by  Wallis  in  1686. 

The  Symbols  of  Inequality  (>  and  <)  are  evidently  modifica- 
tions of  the  sign  of  equality.  If  parallel  lines  denote  equality, 
oblique  lines  would  naturally  be  used  to  denote  inequality,  the 
lines  converging  toward  the  less  quantity.  They  are  said  to  have 
been  introduced  by  Harriot  in  1651. 

I  have  now  presented,  in  a  connected  and  systematic  manner, 
about  all  that  is  known  concerning  the  origin  of  the  ordinary 
arithmetical  symbols.  All  of  them  belong  to  the  period  of  mod- 
ern history  and  are  the  products  of  the  revival  of  learning.  One 
each  of  the  signs  of  operation  was  furnished  by  France,  England 
and  the  Netherlands,  and  three  by  Germany.  Of  the  other  sym- 
bols named,  all  were  introduced  by  Englishmen,  with  the  excep- 
tion of  the  vinculum,  which  is  due  to  Vieta,  a  Frenchman,  and 
the  parenthesis  and  brackets,  which  were  invented  by  the  Dutch 
mathematician  Girard. 


CHAPTER  IV. 

THE   BASIS   OF  THE   SCALE   OF  NUMEKATION. 

THE  Basis  of  our  scale  of  numeration  and  notation  is  dec- 
imal. This  basis  is  not  essential,  but  accidental.  Man- 
kind commenced  reckoning  by  counting  the  fingers  of  the  left 
hand,  including  the  thumb,  and  thus  at  first  probably  reckoned 
by  fives.  As  the  art  of  numbering  advanced,  they  adopted  a 
group,  derived  from  the  fingers  of  both  hands,  and  thus  ten 
became  the  basis  of  numbering.  The  decimal  base  was  con- 
sequently determined  by  the  number  of  fingers  on  each  hand. 
Had  there  been  three  fingers  and  a  thumb,  the  scale  would 
have  been  octary;  had  there  been  five  fingers  and  a  thumb, 
the  scale  would  have  been  duodecimal,  which  would  have  been 
a  great  advantage  to  arithmetic,  whatever  it  might  have  been 
to  the  hand  itself. 

The  universal  use  among  civilized  nations  of  the  decimal 
scale  of  numeration  seems  to  imply  some  peculiar  excellence  in 
it.  It  appears  as  if  nature  had  pointed  directly  to  it,  on 
account  of  some  essential  fitness  of  the  number  ten,  as  the 
numerical  basis.  Indeed,  this  opinion  has  been  quite  general, 
and  the  habit  acquired  from  the  use  of  the  system  has  served 
to  confirm  the  belief.  Many  persons  get  the  base  of  numera- 
tion and  the  mode  of  notation  so  mingled  together  that  they 
see  in  the  Arabic  system  nothing  save  the  decimal  basis  of 
numeration,  and  attribute  to  it  all  those  high  qualities  which 
belong  to  the  mode  only.  It  is  this  which  has  led  some  per- 
sons to  regard  the  decimal  basis  as  the  perfection  of  simplicity 
and  utility. 

8  (113) 


114  THE   PHILOSOPHY   OF   ARITHMETIC. 

A  little  reflection,  however,  will  prove  that  such  an  assump- 
tion is  groundless.  Although  the  decimal  scale  has  been 
adopted  by  every  civilized  nation,  yet,  as  has  been  shown,  the 
selection  was  accidental,  and  the  base  entirely  arbitrary.  The 
selection  occurred  before  attention  was  given  to  a  general  sys- 
tem, in  short,  without  reflection,  and  its  supposed  perfection  is 
a  mere  delusion.  Any  other  number  might  have  been  taken 
as  the  root  of  the  numerical  scale ;  and,  were  a  new  basis  to 
be  selected  by  mathematicians  familiar  with  the  properties  of 
numbers,  there  are  several  considerations  that  would  lead  them 
to  adopt  some  other  basis  than  the  decimal.  Some  of  the 
objections  to  the  decimal  basis  will  be  stated,  and  a  few  consid- 
erations presented  in  favor  of  some  other  number  as  the  basis 
of  the  language  of  arithmetic. 

First,  the  decimal  scale  is  unnatural.  It  has  been  super- 
ficially urged  that  the  decimal  scale  is  the  most  natural  one 
that  could  have  been  selected.  On  the  contrary,  there  is  no- 
thing natural  about  it,  except  the  fingers,  and  a  little  reflection 
would  have  shown  that  these  are  grouped  by  fours  instead  of 
fives.  In  fact,  a  group  by  tens  is  seldom  seen,  either  in  nature 
or  in  art.  What  things  exist  by  tens,  associate  by  tens,  or 
separate  into  tenths?  Nature  groups  in  pairs,  in  threes, 
in  fours,  in  fives,  and  in  sixes;  but  seldom,  if  ever,  in 
tens.  Man  doubles  and  triples  and  quadruples  his  units ;  he 
divides  them  into  halves  and  thirds  and  quarters;  but  where 
does  he  estimate  by  tens  or  tenths?  It  is  thus  seen  that  the 
grouping  by  tens  is  an  unnatural  method,  suggested  neither 
by  nature  nor  the  practical  requirements  of  art. 

Second,  the  decimal  scale  is  unscientific.  The  confused 
idea  of  the  relation  of  the  base  of  the  scale  to  the  mode  of 
notation,  has  led  some  to  suppose  that  the  decimal  scale  is  one 
of  the  triumphs  of  science.  The  truth  is,  as  has  already  been 
shown,  that  not  only  was  it  not  established  upon  scientific 
principles,  but  it  is  really  a  violation  of  those  principles.  The 
decimal  scale  originated  by  chance,  bv  a  mere  accident     Men 


THE    BASIS    OF    THE    SCALE    OF    NUMERATION.  115 

had  ten  fingers,  including  the  thumbs,  and  found  it  convenient 
to  reckon  by  counting  their  fingers;  and  thus  acquired  the 
habit  of  counting  by  tens.  Had  science,  instead  of  chance, 
presided  at  its  birth,  we  should  have  a  basis  that  would  have 
given  a  new  beauty  and  a  greater  simplicity  to  our  already 
admirable  system  of  arithmetical  language. 

Third,  the  decimal  scaie  is  also  inconvenient.  It  has  been 
held  not  only  that  the  decimal  basis  is  scientific,  but  that 
it  is  the  most  convenient  one  that  could  have  been  selected. 
It  needs  but  little  reflection  to  see  the  incorrectness  of  this 
assumption.  One  essential  for  the  basis  of  a  scale  is  the 
property  of  its  being  divisible  into  a  number  of  simple  parts, 
so  that  it  may  be  a  multiple  of  several  of  the  smaller  numbers. 
The  number  ten  will  admit  of  only  two  such  divisions,  the 
half  and  the  fifth.  The  third,  fourth,  and  sixth  are  not  exact 
parts  of  the  denary  base,  in  consequence  of  which  it  is  incon- 
venient to  express  these  fractions  in  the  scale.  Were  the 
basis  twelve  instead  of  ten,  we  could  obtain  the  half  third, 
fourth,  and  sixth,  and  these  fractions  could  be  expressed  by 
the  scale  in  a  single  place ;  whereas  the  fourth  now  requires 
two  places  (.25),  and  the  third  and  sixth  cannot  be  expressed 
exactly  in  a  decimal  scale,  except  as  a  circulate. 

Essentials  of  a  Base.— It  will  be  interesting  to  notice  some 
of  the  essentials  of  a  base,  and  to  observe  what  number  com- 
plies most  fully  with  these  requirements.  The  first  essential 
of  a  good  base  is  that  it  will  admit  of  being  divided  into  the 
simple  fractional  parts  ;  the  second  is  that  the  number  be  neither 
too  large  nor  too  small.  The  advantage  of  the  capability  of 
being  divided  into  simple  fractional  parts  is  that  such  fractions 
may  be  readily  expressed  in  the  terms  of  the  scale  as  we  now 
express  decimal  fractions.  In  the  decimal  scale  only  one-half 
and  one-fifth  can  be  expressed  in  one  place  of  decimals,  since 
they  are  the  only  exact  parts  of  ten.  With  a  scale  whose  basis  is 
a  multiple  of  two,  three,  four  and  six,  each  corresponding  frac- 
tion  could  be  expressed  in  terms  of  the  scale  in  a  single  place. 


116  THE   PHILOSOPHY    OF   ARITHMETIC. 

In  respect  to  the  size  of  the  base,  if  the  number  is  too 
small,  it  will  require  too  many  names  and  places  to  express 
large  numbers.  If  the  number  is  very  large,  it  will  group 
together  too  many  units  to  be  apprehended  and  easily  used  in 
numerical  operations. 

Other  Scales. — There  are  several  other  bases  which  have  been 
recommended  as  preferable  to  the  decimal ;  the  most  important 
of  which  are  the  Binary,  the  Octary,  and  the  Duodecimal.  The 
Binary  scale  was  proposed  and  strongly  advocated  by  Leibnitz. 
He  maintained  that  it  was  the  most  natural  method  of  counting, 
and  that  it  presented  great  practical  and  scientific  advantages. 
He  even  constructed  an  arithmetic  upon  this  basis,  called  Binary 
Arithmetic.  The  obvious  objection  to  this  base  is,  that  it 
would  require  too  many  names  and  too  many  places  in  writing 
large  numbers.  The  Octary  system  has  also  been  strongly 
advocated.  A  very  able  article  in  an  American  journal  says 
that  the  binary  base  is  the  only  proper  base  for  gradation,  and 
the  octary  is  the  true  commercial  base  of  numeration  and  nota- 
tion. 

It  is  probable,  however,  taking  all  things  into  consideration, 
that  the  duodecimal  scale  would  be  the  most  suitable.  The 
number  twelve  is  neither  too  large  nor  too  small  for  conveni- 
ence. Its  susceptibility  of  division  into  halves,  thirds,  fourths, 
and  sixths,  is  an  especial  recommendation  to  it.  So  great  are 
these  advantages,  that,  if  the  base  were  to  be  changed,  the 
duodecimal  base  would,  without  doubt,  be  selected. 

The  advantage  of  the  duodecimal  scale  is  especially  apparent 
in  the  expression  of  fractions  in  a  form  similar  to  our  decimal 
fractions.  In  the  decimal  scale,  J  and  \  are  the  only  simple 
fractions  that  can  be  expressed  by  the  scale  in  a  single  place; 
A  cannot  be  expressed  at  all  as  a  simple  decimal ;  J  requires 
two  places,  and  -*-,  like  ^,  gives  an  interminate  decimal.  With 
a  duodecimal  scale  we  could  express  J,  ^,  \,  and  J  in  a 
single  place;  while  ^  and  \  would  require  only  two  places. 
Thus,  in  the  duodecimal  scale,  we  should  have  ^=.6;  s=.4; 


THE   BASIS    OF   THE    SCALE   OF   NUMERATION.  117 

4=-35  ^=.2;  J=.16,  and  i=.14.  This  is  a  very  great  sim- 
pliGcation;  and  since  all  combinations  of  2  and  3  could  be 
readily  expressed,  and  since  these  constitute  such  a  large  pro- 
portion of  numbers,  it  is  evident  that  the  simplification  of  the 
subject,  by  means  of  a  duodecimal  scale,  would  be  very  con- 
siderable. 

I  will  arrange  the  expressions  of  these  fractions  in  the  deci- 
mal and  duodecimal  scales,  side  by  side,  that  the  advantage  of 
the  latter  may  be  more  clearly  seen. 

Decimal  Scale.  Duodecimal  Scale. 


i=.5 

£=•166+ 

i=.& 

i=-2 

i=.333+ 

|=.14285T 

*=•* 

+=.186*35 

i=.25 

£=•125 

i=.3 

4=16 

5        •*J 

i=.lll+ 

i=.249'7 

i=.u 

It  will  be  seen  that  in  the  decimal  scale  all  the  simple  frac- 
tions used  in  practice,  except  J,  give  circulates  or  require  two 
or  three  figures  to  express  them;  while  in  the  duodecimal  scale 
all  the  fractions  ordinarily  used  in  business  transactions  are 
expressed  in  a  single  place,  and  even  J  and  J  require  only  two 
places.  The  fractions  J-  and  \  cannot  be  exactly  expressed  in 
the  scale,  but  these  fractions  are  seldom  used  in  business.  It 
will  be  interesting  to  notice  that  \  and  -f  both  give  perfect  rep- 
etends  in  the  duodecimal  scale,  and  that  they  possess  the  same 
properties  as  perfect  repetends  in  the  decimal  scale. 

There  seems  to  have  been  a  natural  tendency  towards  a  duo- 
decimal scale.  Thus,  a  large  number  of  things  are  reckoned 
by  the  dozen,  and  this  scale  is  even  extended  to  the  gross  and 
the  great-gross  ;  that  is,  to  the  second  and  the  third  powers  of 
the  base.  Again,  in  our  naming  of  numbers,  the  terms  eleven 
and  twelve  seem  to  postpone  the  forming  of  a  group  until  we 
reach  a  dozen.  A  similar  fact  appears  in  extending  the  multi- 
plication table  to  include  twelve  times,  since,  with  the  deci- 
mal scale,  it  could  conveniently  stop  with  nine  or  ten  times. 
The  division  of  the  year  into  twelve  months,  the  circle  into 
twelve  signs,  the  foot  into  twelve  inches,  the  pound  into  twelve 


118  THE   PHILOSOPHY    OF    ARITHMETIC. 

ounces,  etc.,  are  each  a  further  indication  of  the  same  ten- 
dency. 

Change  of  Base. — The  objections  to  the  decimal  base  have 
led  scientific  men  to  advocate  a  change  in  our  scale  of  numer- 
ation and  notation.  Such  a  change  would,  without  doubt,  be 
a  great  advantage,  both  to  science  and  to  art;  yet  the  practi- 
cal difficulties  attending  such  a  change  are  so  great  that  it 
seems  to  be  almost  impossible.  A  change  in  the  base  would 
require  a  complete  change  in  the  oral  language  of  arithmetic. 
The  decimal  scale  is  so  interwoven  with  the  speech  of  nations, 
that  such  a  change  could  be  effected  only  after  years  of 
labor.  For  a  while,  it  would  be  necessary  to  have  two  methods 
of  arithmetic  taught  and  in  use,  as  in  Europe  at  the  time  of 
the  transition  from  the  Roman  to  the  Arabic  system  of  nota- 
tion. The  learned  would  soon  adopt  the  new  method,  but  the 
common  people  would  cling  with  such  tenacity  to  the  old,  that 
even  a  century  might  intervene  before  the  new  method  would 
become  generally  established. 

Will  this  change  ever  be  made  ?  is  a  question  which  is 
sometimes  asked.  I  do  not  know  ;  but  I  am  strongly  in  favor 
of  it,  and  believe  it  possible.  The  diffusion  of  popular  educa- 
tion will  prepare  the  way  for  it,  by  removing  the  difficulties  of 
its  adoption.  These  difficulties,  though  great,  are  not  insur- 
mountable. Changes  of  notation  have  taken  place  in  several 
different  nations,  and  some  nations  have  changed  two  or  three 
times.  The  Greeks  changed  theirs,  first  for  the  alphabetic, 
and  afterwards,  with  the  rest  of  the  civilized  world,  for  the 
Arabic  system.  The  Arabs  themselves  first  adopted  the  Greek, 
and  afterwards  changed  it  for  the  Hindoo  method.  The  peo- 
ple of  Europe  changed  from  the  Roman  to  the  Arabic  system 
even  as  late  as  the  fourteenth  century,  though  it  took  one  01 
two  centuries  to  effect  the  transition.  What  was  done  thus 
early  in  the  history  of  science,  could,  with  the  increased  intel- 
ligence of  our  people,  be  much  more  readily  accomplished  at 
the  present  day.     A  writer  in  one  of  our  American  periodicals 


THE   BASIS   OF   THE   SCALE   OF   NUMERATION.  119 

says:  "The  probability  is  that  it  will  be  done.  The  question 
is  one  of  time  rather  than  of  fact,  and  there  is  plenty  of  time. 
The  diffusion  of  education  will  ultimately  cause  it  to  be  de- 
manded." 

It  is  a  curious  fact,  and  one  worthy  of  remembrance,  that 
Charles  XII.  of  Sweden,  a  short  time  before  his  death,  while 
lying  in  the  trenches  before  the  Norwegian  fortress  of  Freder- 
ickshall,  seriously  deliberated  on  a  scheme  of  introducing  the 
duodecimal  system  of  numeration  into  his  dominions. 


CHAPTER  V. 

OTHER  SCALES   OP  NUMERATION. 

AS  wo  have  seen,  any  number  might  have  been  taken  as  the 
basis  of  the  scale  of  numeration,  the  number  ten,  the 
basis  of  our  present  scale,  being  selected  from  the  circumstance 
of  there  being  ten  fingers  on  the  two  hands.  Some  other  scales 
have  actually  existed,  and  it  will  be  interesting  to  notice,  in 
various  languages,  traces  of  an  earlier  and  simpler  mode  of 
reckoning.  In  order  to  a  clearer  notion  of  the  subject,  it  may 
be  premised  that  a  scale  whose  basis  is  two  is  called  Binary  ; 
three,  Ternary  ;  four,  Quaternary ;  five,  Quinary;  six,  Senary; 
seven,  Septenary  ;  eight,  Octary  ;  nine,  Nonary  ;  ten,  Denary; 
twelve,  Duodenary,  etc. 

The  earliest  method  of  numeration  was  that  of  combining 
units  in  pairs.  It  is  still  familiar  among  sportsmen,  who 
reckon  by  braces  or  couples.  Some  feeble  traces  of  the  Binary 
system  are  found  in  the  early  monuments  of  China.  Fouhi, 
the  founder  and  first  emperor  of  that  vast  monarchy,  is  vener- 
ated in  the  East  as  a  promoter  of  geometry  and  the  inventor 
of  a  science,  the  knowledge  of  which  has  been  lost.  The  em- 
blem of  this  occult  science  appears  to  consist  of  eight  separate 
clusters  of  three  parallel  lines  or  trigrams,  drawn  one  above 
the  other  after  the  Chinese  manner  of  writing,  and  represented 
either  as  entire  or  broken  in  the  middle.  These  varied  tri- 
grams were  called  Eoua  or  suspended  symbols,  from  the  cus- 
tom of  hanging  them  up  in  the  public  places.  In  the  formation 
of  such  clusters,  we  may  perceive  the  application  of  the  binary 

(120) 


OTHER   SCALES   OF   NUMERATION.  121 

scale  as  far  as  three  ranks,  or  the  number  eight.  The  entire 
lines  are  supposed  to  signify  one,  two,  or  four,  according  to 
their  order,  while  the  broken  lines  are  valueless,  and  servo 
merely  to  indicate  the  rank  of  the  others.  If  this  be  true,  it 
furnishes  an  example  of  a  species  of  arithmetic  with  the 
device  of  place,  possessing  an  antiquity  of  more  than  3000 
years. 

The  Binary  scale,  though  never  fully  adopted  by  any  nation 
as  a  method  of  counting,  has  been  recommended  by  one  of  the 
most  celebrated  modern  philosophers,  Leibnitz,  as  presenting 
many  advantages,  from  its  enabling  us  to  perform  all  the 
operations  in  symbolic  arithmetic  by  mere  addition  and  sub- 
traction. Such  a  system  would,  of  course,  require  but  two 
symbols,  unity  and  zero,  by  means  of  which  all  numbers  could 
be  expressed.  Thus,  two  would  be  expressed  by  10,  three  by 
11,  four  by  100,  five  by  101,  six  by  110,  seven  by  111,  eight 
by  1000,  etc.  This  system  was  studiously  circulated  by  its 
author  by  means  of  scientific  journals  and  his  extensive  cor- 
respondence; and  was  communicated  by  him  to  Bouvet,  a 
Jesuit  missionary  at  Pekin,  at  that  time  engaged  in  the  study 
of  Chinese  ambiguities,  and  who  imagined  that  he  had  discov- 
ered in  it  a  key  to  the  explanation  of  the  Cova,  or  lineations 
previously  referred  to. 

This  system  was  also  recommended  by  the  theological  idea 
associated  with  it,  of  which  it  was  claimed  to  be  the  represent- 
ative. As  unity  was  considered  the  symbol  of  Deity,  the  forma- 
tion of  all  numbers  out  of  zero  and  unity  was  considered,  in 
that  age  of  metaphysical  dreaming,  as  an  apt  image  of  the  crea- 
tion of  the  world,  by  God,  from  chaos.  It  was  with  reference  to 
this  view  of  the  binary  arithmetic,  that  a  medal  was  struck,  bear- 
ing on  its  obverse,  as  an  inscription,  the  Pythagorean  distich, 
Numero  Deus  impari  gaudet,  and  on  its  reverse  the  appropri- 
ate verse  descriptive  of  the  system  which  it  celebrated,  Omnibus 
ex  nihilo  ducendis  sufficit  Unum.  The  good  Jesuit,  who 
seemed  to  have  caught  the  spirit  of  Chinese  belief,  regarded 


122  THE   PHILOSOPHY   OE   ARITHMETIC. 

the  Cova,  which  were  supposed  to  conceal  great  mysteries,  as 
the  symbols  of  binary  arithmetic,  as  a  most  mysterious  testi- 
mony to  the  unity  of  the  Deity,  and  as  containing  within  them 
the  germ  of  all  the  sciences. 

To  count  by  threes  was  another  step,  and  this  has  been  pre- 
served by  sportsmen  under  the  term  leash,  meaning  the  strings 
by  which  three  dogs,  aud  no  more,  can  be  held  at  once  in  the 
hand.  The  numbering  by  fours  has  had  a  more  extensive 
application;  it  was  evidently  suggested  by  the  custom  of  tak- 
ing, in  the  rapid  counting  of  objects,  a  pair  in  each  hand,  and 
thus  reckoning  by  fours.  English  fishermen,  who  generally 
count  in  this  way,  call  every  double  pair  (of  herring,  for 
instance),  a  throw  or  cast ;  and  the  term  warp,  which  origin- 
ally meant  to  throw,  is  employed  to  denote  four,  in  various 
articles  of  trade.  It  is  alleged  that  the  Guaranis  and  Sulos, 
two  of  the  lowest  races  of  savages  inhabiting  the  forests  of 
South  America,  count  only  by  fours  ;  at  least  they  express  the 
number  five  by  four  and  one,  six  by  four  and  two,  seven  by 
four  and  three,  etc.  It  has  been  inferred,  also,  from  a  passage 
in  Aristotle,  that  a  certain  tribe  of  Thracians  were  accustomed 
to  use  the  quaternary  scale  of  numeration. 

The  Quinary  system,  which  reckons  by  fives,  or  pentads, 
has  its  foundation  in  the  practice  of  counting  the  fingers  of 
one  hand.  It  appears,  from  the  statements  of  travellers,  to 
have  been  adopted  by  various  savage  nations.  Thus,  certain 
tribes  of  South  America  were  found  to  reckon  by  fives,  which 
they  called  hands.  In  counting  six,  seven,  and  eight,  they 
added  to  the  word  hand  the  names  one,  two,  three,  etc.  Mungo 
Park  found  that  the  same  system  was  practiced  by  the  Yolofs 
and  Foulahs  of  Africa,  who  designate  ten  by  two  hands,  fifteen 
by  three  hands,  etc.  The  quinary  system  seems  also  to  have 
been  formerly  used  in  Persia;  the  word  pende,  which  denotes 
five,  having  the  same  derivation  as  pentcha,  which  signifies 
a  hand.  It  is  even  partially  used  in  England  among  whole- 
sale traders.     In  reckoning  articles  delivered  at  the  warehouse, 


OTHER    SCALES   OF   NUMERATION.  123 

the  person  who  takes  charge  of  the  tale,  having  traced  a  long 
horizontal  line,  continues  to  draw,  alternately  above  and  below 
it,  a  warp,  or  four  vertical  strokes,  each  set  of  which  he  crosses 
by  an  oblique  score,  and  calls  out  tally  as  often  as  the  number 
five  is  completed.  This  custom  is  a  very  general  one  in 
assemblies  where  votes  are  counted,  and  in  similar  circum- 
stances elsewhere. 

The  Senary  method,  so  far  as  we  can  learn,  was  never  used 
by  any  tribe  or  nation;  at  least  never  arose  spontaneously. 
It  is  said  to  have  been  adopted  at  one  time  in  China  by  the 
order  of  a  capricious  tyrant,  who,  having  conceived  an  astro- 
logical fancy  for  the  number  six,  commanded  its  several  combi- 
nations to  be  used  in  all  concerns  of  business  or  learning 
throughout  his  vast  empire. 

The  Septenary  scale  has  not,  so  far  as  we  can  learn,  been 
used  anywhere.  The  number  seven  has  been  regarded  as  a 
kind  of  magic  number,  but  nothing  in  nature  suggested  the 
method  of  counting  by  sevens.  The  division  of  the  year  into 
periods  consisting  of  seven  days  each,  a  custom  among  nearly 
all  nations,  has  given  the  number  seven  a  wide  distinction,  and 
its  frequent  use  in  the  Bible  has  caused  it  to  be  regarded  as 
a  sacred  number,  the  basis  of  a  celestial  system  of  reckoning. 
The  Octary  scale,  also,  though  it  would  possess  many  advant- 
ages, and  has  been  recommended  by  scientific  writers,  has 
never  made  its  appearance  in  any  language.  A  Nonary  scale 
has  also  never  been  used,  and  would  be  the  most  inconvenient 
of  the  smaller  scales  except  the  septenary. 

The  Denary  scale  is  the  system  which  has  prevailed  among 
all  civilized  nations,  and  has  been  incorporated  into  the  very 
structure  of  their  language.  This  universal  method  manifests 
the  existence  of  some  common  principle  of  numbering,  which 
was  the  practice,  so  familiar  in  the  earlier  periods  of  society, 
of  reckoning  by  counting  the  fingers  on  both  hands.  The 
origin  of  the  terms  used  in  the  more  polished  ancient  languages 
is  not  easily  traced,  but  in  the  roughness  of  savage  dialects 


124  THE    PHILOSOPHY    OF    ARITHMETIC. 

these  names  vary  less  from  the  primitive  words.  The  Muysca 
Indians  were  accustomed  to  reckon  as  far  as  ten,  which  they 
called  quihicha  or  a  foot,  referring,  no  doubt,  to  the  number  of 
toes  on  their  bare  feet ;  and  beyond  this  number  they  used 
terms  equivalent  to  foot  one,  foot  tivo,  etc.,  for  eleven,  twelve, 
etc.  Another  South  American  tribe  called  ten,  tunca,  and 
merely  repeated  the  word  to  signify  a  hundred,  or  a  thousand, 
thus :  tunca-tunca,  tunca-tunca-tunca.  The  Peruvian  language 
was  actually  richer  in  the  names  of  numerals  than  the  Greek 
or  Latin.  The  Romans  went  no  higher  than  mille,  a  thou- 
sand, and  the  Greeks  than  /ivpta,  or  ten  thousand.  But  the 
Peruvians  had  the  expressions,  hue,  one  ;  chunca,  ten ;  pachac, 
a  hundred;  huaranca,  a  thousand;  and  hunu,  a  million.  It 
appears  from  an  early  document,  that  the  Indian  tribes  of  New 
England  used  the  Denary  scale,  and  had  distinct  words  to  ex- 
press the  numbers  as  far  as  a  thousand.  The  Laplanders  join 
the  cardinal  to  the  ordinal  numbers ;  thus,  for  eleven  they  say 
auft  nubbe  lokkai,  that  is,  one  to  the  second  ten.  The  origin 
of  the  numerals  in  our  own  dialect  will  be  found  treated  at 
greater  length  in  another  place. 

The  mode  of  reckoning  by  twelves  or  dozens,  may  be  sup- 
posed to  have  had  its  origin  in  the  observation  of  the  celestial 
phenomena,  there  being  twelve  months  or  lunations  commonly 
reckoned  in  a  solar  year.  The  Romans  likewise  adopted  the 
same  number  to  mark  the  subdivisions  of  their  unit  of  measure 
or  of  weight.  The  scale  appears  also  in  our  subdivisions  of 
weights  and  measures,  as  twelve  ounces  to  a  pound,  twelve 
inches  to  a  foot;  and  is  still  very  generally  employed  in 
wholesale  business,  extending  to  the  second  and  even  to  the 
third  term  of  the  progression.  Thus,  twelve  dozen,  or  144, 
make  the  long  hundred  of  the  northern  nations,  or  the  gross 
of  traders;  and  twelve  times  this  again,  or  1128,  make  the 
double  or  great  gross. 

The  scale  of  numeration  by  twenties  has  its  foundation  in 
nature,  like  the  quinary  and  denary.  In  a  rude  state  of 
society,  before  the  discovery  of  other  methods  of  numeration, 


OTHER    SCALES    OF   NUMERATION.  125 

men  might  avail  themselves  for  this  purpose,  not  merely  of  the 
fingers  on  the  hands,  but  also  of  the  toes  on  the  naked  feet ; 
and  such  a  practice  would  naturally  lead  to  the  formation  of  a 
vicenary  scale  of  numeration.  The  languages  of  many  tribes 
indicate  this  method,  and  many  savage  tribes  do  thus  actually 
reckon.  It  is  said  of  the  inhabitants  of  the  peninsula  of  Kam- 
schatka,  that  "it  is  very  amusing  to  see  them  attempt  to  reckon 
above  ten ;  for  having  reckoned  the  fingers  of  both  hands,  they 
clasp  them  together,  which  signifies  ten;  then  they  begin  at  their 
toes  and  count  twenty,  after  which  they  are  quite  confused  and 
cry  matcha,  where  shall  I  take  more?"  Among  the  Caribbees 
who  constituted  the  native  population  of  Barbadoes  and  other 
islands  of  the  Caribbean  sea,  the  numeration  beyond  five  was 
carried  on  by  means  of  the  fingers  and  toes,  and  their  numer- 
ical language  became  generally  descriptive  of  their  practical 
method  of  counting.  The  Abipones,  an  equestrian  people  of 
Paraguay,  to  express  five  show  the  fingers  of  one  hand;  to 
express  ten,  the  fingers  of  both  hands;  "for  twenty,  their 
expression  is  pleasant,  being  allowed  to  show  all  the  fingers  of 
their  hands  and  the  toes  of  their  feet." 

Traces  of  reckoning  by  scores  or  twenties,  are  found  in  our 
own  and  other  European  idioms.  The  expression  threescore 
and  ten  is  familiar.  The  term  score  itself,  which  originally 
meant  a  notch  or  incision  made  on  a  tally  to  signify  the  suc- 
cessive completion  of  such  a  number,  seems  to  indicate  that 
such  a  mode  of  counting  was  most  familiarly  used  by  our  ances- 
tors. The  vicenary  scale  seems  to  have  prevailed  very  exten- 
sively among  the  Scandinavian  nations,  as  is  shown  by  the 
vestiges  of  it  both  among  them  and  the  languages  partly 
derived  from  them.  The  French  language  has  no  term  for  the 
numbers  in  the  second  series  of  the  denary  scale  above  soix- 
ante  or  sixty.  Eighty  is  expressed  by  quatre-vingts,  four 
twenties,  and  ninety  by  quatre-vingts-dix,  four  twenties  and 
ten.  The  people  of  Biscay  and  Armorica  are  said  still  to 
reckon  by  the  powers  of  twenty,  and,  according  to  Humboldt, 
the  same  mode  of  numeration  was  employed  by  the  Mexicans. 


CHAPTER  VI. 
A  DUODECIMAL   SCALE. 

AS  already  explained,  any  number  may  be  made  the  basis 
of  a  system  of  numeration  and  notation.  The  decimal 
basis  is  a  mere  accident,  and  in  some  respects  an  unfortunate 
one,  both  for  science  and  art.  The  duodecimal  basis  would 
have  been  greatly  superior,  giving  greater  simplicity  to  the 
science,  and  facilitating  its  various  applications.  In  this 
chapter  it  will  be  explained  how  arithmetic  might  have  been 
developed  upon  a  duodecimal  basis. 

In  order  to  make  the  matter  clear,  I  call  attention  to  two  or 
three  principles  of  numeration  and  notation.  First,  the  bases 
of  numeration  and  notation  should  be  the  same  ;  that  is,  if  we 
write  numbers  in  a  duodecimal  system,  we  should  also  name 
numbers  by  a  duodecimal  system.  Second,  in  naming  num- 
bers by  any  system,  we  give  independent  names  up  to  the  base, 
and  then  reckon  by  groups,  using  the  simple  names  to  number 
the  groups.  Bearing  these  principles  in  mind,  we  are  ready 
to  understand  Numeration,  Notation,  and  the  Fundamental 
Rules  in  Duodecimal  Arithmetic. 

Numeration. — In  naming  numbers  by  the  duodecimal 
system,  we  would  first  name  the  simple  numbers  from  one  to 
eleven,  and  then,  adding  one  more  unit,  form  a  group,  and  name 
this  group  twelve.  We  would  then,  as  in  the  decimal  system,  use 
these  first  names  to  number  the  groups.  Naming  numbers  in 
this  way,  we  would  have  the  simple  names,  one,  two,  three,  etc., 
up  to  twelve.  Continuing  from  twelve,  we  would  have  one  and 
twelve,  two  and  twelve,  three  and  twelve,  etc.,  up  to  twelve  and 
twelve,  which  we  would  call  two  twelves.     Passing  on  from  this 

(126) 


A    DUODECIMAL   SCALE.  12/ 

we  would  have  two  twelves  and  one,  two  twelves  and  two,  etc., 
to  three  twelves,  and  so  on  until  we  reach  twelve  twelves,  when 
we  would  form  a  new  group  containing  twelve  twelves,  and 
give  this  new  group  a  new  name,  as  gross,  and  then  employ 
the  first  simple  names  again  to  number  the  gross.  In  this  way 
we  would  continue  grouping  by  twelves,  and  giving  a  new 
name  to  each  group,  as  in  the  decimal  scale  by  tens,  as  far  as 
is  necessary. 

These  names,  in  the  duodecimal  system,  would  naturally 
become  abbreviated  by  use,  as  the  corresponding  names  in  the 
decimal  system.  Thus,  as  in  the  decimal  system  ten  was 
changed  to  teen,  we  may  suppose  twelve  to  be  changed  to  teel, 
and  omitting  the  "  and"  as  in  the  common  system,  we  would 
count  one-teel,  two-teel,  thir-teel,  four-teel,  fif-teel,  six-teel,  etc., 
up  to  eleven-teel.  Two-twelves  might  be  changed  into  two-tel, 
or  twen-tel,  corresponding  to  two-ty  or  twenty,  and  we  would 
continue  to  count  twentel-one,  twentel-two,  etc.  Three-twelves 
might  be  contracted  into  three-tel  or  thirtel,  corresponding  to 
three-ty  or  thirty  of  the  decimal  system;  four-twelves  to 
fourtel,  five  twelves  to  fiftel,  etc.,  up  to  a  gross.  Proceeding 
in  the  same  manner,  a  collection  of  twelve  gross  would  need 
a  new  name,  and  thus  on  to  the  higher  groups  of  the  scale. 

In  this  manner,  the  names  of  numbers  according  to  a  duo- 
decimal system  could  be  easily  applied.  Were  we  actually 
forming  such  a  system,  the  simplest  method  would  be  to  intro- 
duce only  a  few  new  names  for  the  smaller  groups,  and  then 
take  the  names  of  the  higher  groups  of  the  decimal  system, 
with  perhaps  a  slight  modification  in  their  orthography  and 
pronunciation,  to  name  the  higher  groups  of  the  new  scale. 
Thus,  million,  billion,  etc.,  could  be  used  to  name  the  new 
groups  without  any  confusion,  as  they  do  not  indicate  any 
definite  number  of  units  to  the  mind,  but  merely  so  many  col- 
lections of  smaller  collections.  Indeed,  even  the  word  thou- 
sand, with  a  modification  of  its  orthography,  say  ihousun, 
might  be   used  to  represent  a  collection   of   twelve  groups, 


128  THE   PHILOSOPHY   OF   ARITHMETIC. 

each  containing  a  gross,  without  any  confusion  of  ideas. 
Their  etymological  formation  would  not  be  an  objection  of  any 
particular  force,  as  no  one  in  using  them  thinks  of  their  pri- 
mary signification.  These  terms  are  not  suggested  as  the 
best,  but  as  the  simplest  in  making  the  transition  from  the 
old  to  the  new  system.  It  will  also  be  noticed  that  our 
departure  in  the  decimal  scale  from  the  principle  of  the  sys- 
tem, by  using  the  terms  eleven  and  twelve,  would  facilitate  the 
adoption  of  a  duodecimal  system. 

To  illustrate  the  subject  more  fully,  let  us  adopt  the  names 
suggested,  and  apply  them  to  the  scale.  Naming  numbers 
according  to  the  method  explained,  we  would  have  the  names 
as  indicated  in  the  following  series: 

one 

two 

three 

four 

five 

six 

seven 

eight 

nine 

ten 

eleven 

twelve 

Notation. — The  writing  of  numbers  by  the  duodecimal 
system  would  be  an  immediate  outgrowth  of  the  method  of 
naming  numbers  in  this  system.  As  in  the  decimal  system  of 
notation,  it  would  be  necessary  to  employ  a  number  of  char- 
acters one  less  than  the  number  of  units  in  the  base,  besides 
the  character  for  nothing.  Since  the  group  contains  twelve 
units,  the  number  of  significant  characters  would  be  eleven — 
two  more  than  in  the  decimal  system.  For  these  characters 
we  should  use  the  nine  digits  of  the  decimal  system,  and  then 
introduce  new  characters  for  the  numbers  ten  and  eleven.  To 
illustrate,  we  will  represent  ten  by  the  character  4>  and  eleven 
by  n. 

These  characters,  with  the  zero,  would  be  combined  to  rep- 
resent numbers  in  the  duodecimal  scale  in  the  same  manner  as 
the  nine  digits  represent  numbers  in  the  decimal  scale.     Thus, 


oneteel 

twentel-one 

one  gross  and  one 

twoteel 

twentel-two 

one  gross  and  two 

thirteel 

twentel-eight 

two  gross  and  five 

fou  rt  eel 

twentel-eleven 

six  gross  and  seven 

nfteel 

thirtel-one 

ten  gross  and  eight 
eleven  gross  and  nine 

sixteel 

fortel-two 

seven  teel 

fi  ft  el-six 

one  thousun 

eigliteel 

sixtel-eight 

one  thousun  and  five 

nineteel 

seventel-nine 

one  thousun  four  gross 

lenteel 

tentel-ten 

and  seven 

eleventeel 

eleven  tel-eleven 

two  thousun  seven  gross 

twentel 

one  gross 

and  fortel-one 

A    DUODECIMAL    SCALE. 


129 


twelve  would  be  represented  by  10,  signifying  one  of  the 
groups  containing  twelve;  11  would  represent  one  and  twelve, 
or  oneteel;  12  would  represent  two  and  twelve,  or  twoteel;  13 
would  represent  thirteel;  14,  fourteel;  15,  fifteel,  etc.  Con- 
tinuing thus,  20  would  represent  two  twelves,  or  twentel;  21, 
twentel-one;  23,  twentel-three,  etc.  The  notation  of  numbers 
up  to  a  thousun  may  be  indicated  as  follows : — 


one,  1 

twel 

ve, 

10 

thirtel,  30 

two,  2 

oneteel, 

11 

thirtel-two,  32 

three, 

3 

twoteel, 

12 

thirtel-five,  35 

etc.,    etc. 

twentel, 

20 

thirtel-ten,  3<f> 

nine,  1 

> 

twentel- 

one,  21 

thirtel-eleven. 

,  3n 

ten,  <f> 

twentel-ten,  2^ 

one 

gross,  100 

eleven 

,  n 

twentel-eleven 

,2ll 

one 

thousun, 

1000 

Extending  the  series 

as 

explained 

above,  we  should 

have 

the  following  notation  table : 

:— 

r 

FABLE 

to 

CO 

P 

CO 

p 
p 

p 

QQ 

5 

CO* 

CO 

CO 

P 

P 

p 

S 
o 

a 

S3 

3 

1 

CHI 

O 

P 
P 

CO 

P 
O 

-P 

H 

P 

o 
EH 

cm 

O 

CO 

CO           *,_ 

co 

CO 

«m 

CO 

CO 

5*- 

co 

p 

to 

rillyui 
ross  o 

cp 

a 

S3 

O 

CO 
CO 

o 

M 

CO 

P 

P 

o 

CO 

co 
o 

F* 

CO 

g 

p 
o 
-p 

ross. 
welve 

CO 

*p 

FH      O 

En 

s 

o 

H 

§ 

O 

EH 

H 

O      EH 

p 

8       5 

n 

4 

6 

n 

8 

5 

7 

<S> 

3       6 

5 

From  the  explanation  given  it  is  clearly  seen  that  a  system 
of  duodecimal  arithmetic  might  be  easily  developed,  and  read- 
ily learned  and  reduced  to  practice.  Employing  the  names 
which  I  have  indicated,  or  others  similar  to  them,  the  change 
from  the  decimal  to  the  duodecimal  system  would  be  much 
less  difficult  than  has  usually  been  supposed.  It  would  be 
necessary  to  learn  the  method  of  naming  and  writing  num- 
bers, which  we  have  seen  is  very  simple,  and  a  new  addition 
9 


ISO  THE   PHILOSOPHY   OF    ARITHMETIC. 

and  multiplication  table,  from  which  we  could  readily  derive 
the  elementary  differences  and  quotients.  The  rest  of  the  sci- 
ence would  be  readily  acquired,  as  all  of  its  methods  and 
principles  would  remain  unchanged.  Indeed,  so  readily  could 
the  change  be  made,  that  in  view  of  the  great  advantages  of 
the  system,  one  is  almost  ready  to  believe  that  the  time  will 
come  when  scientific  men  will  turn  their  attention  seriously  to 
the  matter  and  endeavor  to  effect  the  change. 

Fundamental  Operations. — In  order  to  show  how  read- 
ily the  transition  could  be  made,  I  will  present  the  method  of 
operation  in  the  fundamental  rules.  We  would  proceed  first 
to  form  an  addition  table  containing  the  elementary  sums, 
which,  as  in  the  decimal  system,  we  would  commit  to  memory. 
From  this  we  could  readily  derive  the  elementary  differences 
used  in  subtraction.     Such  a  table  is  given  on  page   131. 

By  means  of  this  table  we  can  readily  find  the  sum  or  dif- 
ference of  numbers  expressed  in  the  duodecimal  system.     To 
illustrate,  required   the    sum   of    487n,   54  38,     operation. 
63n7,  4>856.     The  solution  of  this  would  be  as  487n 

follows:  Adding  the  column  of  units,  6  units  5$ 38 

and  7  units  are   11  units,  and  8   units  are  19  63ii7 

units,  and  n  units  are  28  units,  or   2   twelves  

23748 
and  8  units ;  writing  the  units,  and  carrying 

2  to  the  column  of  twelves,  we  have  2  twelves  and  5  twelves 

are  7  twelves,  and  n  twelves  are  16  twelves,  and  3  twelves 

are  19  twelves,  and  7  twelves  are  24  twelves,  or  2  gross  and 

4  twelves;    writing  the  twelves,  and  carrying  2  to  the  third 

column,  we  have  2  gross  and  8  gross  are  4>  gross,  and  3  gross 

are  11  gross,  and  $  gross  are  In  gross,  and  8  gross  are  27  gross, 

or  2  thousuns  and  7  gross;  2  thousuns  and  *  thousuns  are  10 

thousuns,  and  6  thousuns  are  16  thousuns,  and  5  thousuns  are 

In   thousuns,   and  4  thousuns   are   23  thousuns;    hence   the 

amount  is  23748. 

To  illustrate  subtraction  let  it  be  required  to  find  the  differ- 


A    DUODECIMAL    SCALE. 


131 


ADDITION    AND   MULTIPLICATION   TABLES   IN   THE   DUODECIMAL   SCALE. 


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132  -the  philosophy  of  arithmetic. 

ence  between  6428  and  2564.    We  would  solve  this  as  follows- 

Subtracting  4  units  from  8  units  we  have  4 

units  remaining;  we  cannot  take   6   twelves     operation. 

from  2  twelves,  so  we  add  10  twelves  and  have  6428 

12  twelves;  6  twelves  from  12  twelves  leaves 

8  twelves;  carrying  1  to  5  we  have  6  gross;  3<i>84 

we  cannot  take  6  gross  from  4  gross;  adding  10  as  before  we 

have  6  gross  from  14  gross  leaves  $  gross;  adding  1  thousun 

to  2  thousuns,  we  have  3  thousuns  from  6  thousuns  leaves  3 

thousuns;  hence  the  remainder  is  3*84. 

In  order  to  multiply  and  divide,  we  first  form  a  multiplica- 
tion table  similar  to  that  now  used  in  the  decimal  system,  and 
commit  it  to  memory.  This  table  need  not  extend  beyond 
"  twelve  times,"  as  in  our  present  system  there  is  no  need  of 
extending  beyond  "ten  times."  From  this  table  of  elementary 
products,  we  can  readily  derive  the  table  of  elementary  quo- 
tients as  we  do  in  the  decimal  system.  Such  a  table  will  be 
found  on  page  131. 

It  will  be  interesting  to  notice  several  peculiarities  of  this 
v,able,  similar  to  those  of  the  decimal  system.  As  the  column 
of  "five  times"  ends  alternately  in  5  and  0,  making  it  so 
easily  learned  by  children,  so  the  column  of  "six  times"  in 
the  duodecimal  table  will  end  alternately  in  6  and  0.  In  our 
present  table  the  sum  of  the  two  terms  of  each  product  in  the 
column  of  "nine  times"  equals  nine,  so  in  the  duodecimal 
table,  the  sum  of  the  two  terms  of  each  product  in  the  column 
of  "eleven  times"  equals  eleven.  We  also  notice  that  each 
product  in  the  column  of  "  twelve  times"  ends  in  0,  as  does 
each  product  in  the  column  of  "ten  times"  of  our  present 
table. 

By  means  of  the  multiplication  table  we  can  readily  find 
the  product  or  quotient  of  numbers  expressed  in  the  duodeci- 
mal scale.  To  illustrate  multiplication,  let  it  be  required  to 
find  the  product  of  54$8  by  3nT.  We  would  solve  this  as 
follows:  Using  the  first  term  of  the  multiplier,  *l  times  8  are 


A  DUODECIMAL   SCALE.  133 

48,  7    times  *  are  5*,  and    4   are    62,  7  operation. 

times  4  are  24  and  6  are  2*,  7  times  5  5^ 

are  2n  and  2  are  31,  making  the  first  __ 

partial  product  31*28;  multiplying  by  4fl5*- 

n  we  have  n  times  8  are  74,  n  times  *  14280 

are  92  and  7  are  99,  n  times  4  are  38  and  ""1953768 

9  are  45,  n  times  5  are  47  and  4  are  411 ; 

3  times  8  are  20,  3  times  *  are  26  and  2  are  28,  3  times  4  ar* 

10  and  2  are  12,  3  times  5  are  13  and  1  are  14.     Adding  up 
the  partial  products,  we  have  as  the  complete  product,  19537 G8. 

To  illustrate  division,  let  it  be  required   to  find  the  quotient 
of  1953768  divided  by  3n7.     We  would  operation. 

solve  this  as  follows  :  We  find  that  the     3117)  1953768(54*8 
divisor   is  contained  in    the   first   four  179n 

terms  of  the  dividend  5  times,  and  mul-  1747 

tiplying  3n7  by  5  we  have  179n  ;  sub-  13*4 

tracting    this    from    the    dividend   we  3636 

have  a  remainder,  174;  bringing  down  337<i> 

the    next   figure   of  the   dividend   and  2788 

proceeding  as  before,  we  have  for  the  " 

quotient  54*8. 

The  method  of  finding  the  square  or  cube  root  of  a  number 
expressed  in  the  duodecimal  scale  is  similar  to  that  used  in 
the  decimal  scale,  as  may  be  shown  by  an  operation 

example.     Thus,  find  the  square  root  of  n-53'01(3  7 

II5301.     The  greatest  square  in  n  is  9;        64)253 
subtracting  and  bringing  down  a  period,  ^14 

and  dividing  by  2  times  3  or  6,  we  find  the        687  v^qi 
second  term  of  the  root  to  be  4;  complet-  3n01 

ing  the  divisor  and  multiplying  64  by  4, 
we  have  214;    subtracting  and  bringing  down,  we  have  3n01, 
and  dividing  by  2  times  34,  or  68,  we  have  7  for  the  last  figure 
of  the  root;  completing  the   divisor  and  multiplying  it  by  7, 
we  have  3n01,  which  leaves  no  remainder. 

The  above  tables  and  calculations  seem  awkward  to  one 


134:  THE    PHILOSOPHY    OF    ARITHMETIC. 

who  is  familiar  with  the  decimal  system ;  but  it  should  be 
remembered  that  a  beginner  would  learn  the  addition  and  mul- 
tiplication tables  and  the  calculations  based  on  them,  just  as 
readily  as  he  now  learns  them  in  the  decimal  system.  The 
practical  value  of  such  a  system,  in  addition  to  what  has 
already  been  said,  may  be  seen  in  the  calculation  of  interest, 
the  rules  for  which  would  be  greatly  simplified  on  account  of 
the  relation  of  the  number  of  months  in  a  year  (12)  to  the 
base,  and  also  of  the  relation  of  the  rate  to  the  same,  which 
would  be  some  8%  or  9%  ;  that  is,  8  or  9  per  gross.  I  hope  to 
be  able  in  a  few  years  to  publish  a  small  work  in  which  the 
whole  science  of  arithmetic  shall  be  developed  on  the  duodeci- 
mal basis. 


CHAPTER  VII. 

GREEK   ARITHMETIC. 

GREEK  Arithmetic,  like  that  of  all  other  nations  of  anti- 
quity, began  in  the  representation  of  numbers  by  strokes  or 
straight  lines.  This  system,  in  the  progress  of  thought  and 
civilization,  was  finally  discarded,  and  the  letters  of  the  alpha- 
bet taken  as  the  symbols  of  numbers.  After  adopting  the 
letters  of  their  alphabet,  the  Greeks  seem  to  have  had  no  less 
than  three  distinct  methods  of  notation.  They  used  the 
letters  in  their  natural  order,  to  signify  the  smaller  ordinal 
numbers.  In  this  way  the  books  of  Homer's  Iliad  and 
Odyssey  are  usually  marked.  They  employed  also  the  first 
letters  of  the  words  for  numerals  as  abbreviated  symbols,  mak- 
ing use  of  an  ingenious  device  for  augmenting  the  powers  of 
these  symbols;  thus,  a  letter  enclosed  by  a  line  on  each  side  and 
another  drawn  over  the  top,  as  Fl,  was  made  to  signify  five 
thousand  times   its  original  value. 

A  more  complete  method  consisted  in  the  distribution  of  the 
twenty -four  letters  of  their  alphabet  into  three  classes,  corre- 
sponding to  units,  tens,  and  hundreds,  adding  another  character 
to  each  class  to  complete  the  symbols  for  all  of  the  nine  digits. 
This  latter  method  was  the  one  in  common  use,  and  that  which 
was  made  the  basis  of  their  arithmetic.  The  units  from  one 
to  nine  inclusive,  were  denoted  by  the  letters  a,  /3, y,  6,  e,  r,  C,  v,Q\ 
the  tens  by  e,  <c,  I,  //,  v,  f,  0,  ?r,  h ;  and  the  hundreds  by  p,  <r,  r,  v,  <f>, 
x,  i>,  u,  7).  Thousands  were  represented  by  the  first  series 
with  the  iota,  or  dash  subscribed,  thus:  a,(3,y,6  etc.  With 
these  characters  they  could  readily  express  any  number  under 

(135) 


13t)  THE    PHILOSOPHY    OF    ARITHMETIC. 

10,000,  or  a  myriad.     Thus,  991  was  expressed  by  7i  ha;  7382, 
by  fnr£;  6420,   by  rwc;  4001,  by  da. 

It  will  be  noticed  that  neither  the  order  nor  the  number  of 
characters  was  considered  in  expressing  numbers.  The  value  of 
the  expression  was  the  same  in  whatever  order  the  letters  were 
placed ;  though  as  regularity  tended  towards  simplicity,  they 
generally  wrote  the  characters  according  to  value,  from  left  to 
right. 

Myriads,  or  ten  thousands,  were  denoted  by  the  letter  m,  a 
letter  representing  the  number  of  myriads  indicated  being 
written  above  it.  Thus,  M  denoted  10,000;  M,  20,000;  £ 
30,000,  etc.  Thus,  also,  ^  denoted  370,000;  ?™0  43720000;  and 
in  general,  the  letter  m  placed  beneath  any  number  had  the 
same  effect  as  our  annexing  four  ciphers. 

This  is  the  notation  employed  by  Eutocius  in  his  commenta- 
ries on  Archimedes,  but  it  is  evidently  inconvenient  in  calcula- 
tion. Diophantus  and  Pappus  expressed  the  myriad  more 
simply  by  the  two  letters  Mv  placed  after  the  number,  and 
afterwards  by  merely  writing  a  point  after  it.  This  enabled 
them  to  express  100,000,000,  which  was  the  greatest  extent  of 
the  ordinary  Greek  arithmetic. 

This  system  had  been  extended  by  Archimedes  and  Apollo- 
nius,  for  the  purpose  of  astronomical  and  other  scientific 
calculations.  Archimedes,  in  order  to  express  the  number 
of  grains  of  sand  that  might  be  contained  in  a  sphere  that  had 
for  its  diameter  the  distance  of  the  fixed  stars  from  the  earth, 
found  it  necessary  to  represent  a  number  which,  with  our  nota 
tion,  would  require  sixty-four  places  of  figures;  and  in  order 
to  do  this,  he  assumed  the  square  myriad,  or  100,000,000,  as  a 
new  unit,  and  the  numbers  formed  with  these  new  units  he 
called  numbers  of  the  second  order ;  and  thus  he  was  enabled 
to  express  any  number  which  in  our  notation  requires  sixteen 
figures.  Assuming  again  100,000, 0002  as  a  new  unit,  he  could 
represent  any  number  that  requires  in  our  scale  twenty-four 


GREEK    ARITHMETIC.  137 

figures,  and  so  on ;  so  that  by  means  of  his  numbers  of  the 
eighth  order,  he  could  express  the  number  in  question,  which 
requires  sixty-four  figures  in  our  scale. 

By  this  system  all  numbers  were  separated  into  periods  or 
orders  of  eight  figures.  This  was  afterwards  considerably 
improved  by  Apollonius,  who,  instead  of  periods  of  eight 
places,  which  were  called  by  Archimedes  octates,  reduced  num- 
bers to  periods  of  four  places ;  the  first  of  which,  on  the  left, 
were  units,  the  second  period  myriads,  the  third  double  myri- 
ads or  numbers  of  the  second  order,  and  so  on  indefinitely. 
In  this  manner  Apollonius  was  able  to  write  any  number  that 
can  be  expressed  by  our  system  of  numeration  ;  as  for  example, 
if  he  had  wished  to  represent  the  circumference  of  a  circle 
whose  diameter  was  a  myriad  of  the  ninth  order,  he  would 
have  written  it  thus: 

y.avie.     Oafr.     y^O.      £. 7)A/3.     yopc.     $XW-     Y^P*     ^v-     ^wk6- 
3.1415  9265  3589     7932     3846    2643    3832    7950    2824 

The  learned  astronomer  Ptolemy  modified  this  system  in  its 
descending  range  by  applying  it  to  the  sexagesimal  subdivisions 
of  the  lines  inscribed  in  a  circle.  He  likewise  advanced  an 
important  step,  by  employing  a  small  or  accentuated  o  to  supply 
the  place  of  any  number  wanting  in  the  order  of  progression. 

The  Greek  method  of  expressing  fractions  was  also  peculiar. 
An  accent  set  on  the  right  of  a  number,  made  of  that  number 
the  denominator  of  a  fraction  whose  numerator  was  a  unit, 
thus,  /=i,  <?'=£,  #'=&,  P>ca'=T!r,  etc.  When  the  numerator  is 
not  unity,  the  denominator  is  placed  as  we  set  our  exponents. 
Thus,  t*&  represented  156*,  or  £f ,  and  fKCl  represented  712\  or 
3-J-p  The  fraction  \  had  a  particular  character,  as  C,  <, 
C,  or  K.  The  notation  of  the  Greeks  was  not  adapted  to  the 
descending  scale,  and  consequently  they  had  no  decimals. 

The  notation  of  the  Greeks,  though  much  inferior  to  that  of 
the  present  day,  was  formed  upon  a  regular  and  scientific 
basis,  and  could  be  employed  with  considerable  convenience 
as  an  instrument  of  calculation.     We  will  present  two  or  three 


138  THE    PHILOSOPHY    OF    ARITHMETIC. 

examples   taken    from   Barlow's    Theory  of  Numbers,   from 
which  some  of  the  previous  facts  are  gathered. 

Addition. — The   following    example   in    addition    is    from 
Eutocius,  Theorem  4,  of  the  Measure  of  the  Circle. 
utf.yfoKa  847   3921 

f.jj*  60  8400 

?\v  .£™a  908  2321 

The  method,  it  will  be  seen,  is  similar  to  compound  addition, 
but  is  simpler  on  account  of  the  constant  ratio  of  ten  between 
any  character  and  the  succeeding  one. 

Subtraction. — The  following  example  in  subtraction  is  from 
Eutocius,  Theorem  3,  on  the  Measure  of  the  Circle. 
d.yxK  93636 

p!yv  d  23409 


C.    a«C  ?0227 

The  method  is  simple,  proceeding  from  right  to  left  as  in 
our  subtraction,  which  seems  so  obviously  advantageous  and 
simple  that  one  can  hardly  conceive  why  the  Greeks  should 
ever  proceed  in  the  contrary  way,  although  there  are  many 
instances  which  make  it  evident  that  they  did,  both  in  addition 
and  subtraction,  work  from  left  to  right. 

Multiplication. — In  multiplication  they  most  commonly  pro- 
ceeded in  their  operations  from  left  to  right,  as  we  do  in  mul- 
tiplication in  algebra,  and  their  successive  products  were 
placed  without  much  apparent  order;  but  as  each  of  their 
characters  retained  always  its  own  proper  value,  in  whatever 
order  they  stood,  the  only  inconvenience  of  this  was,  that  it 
rendered  the  addition  of  them  a  little  more  troublesome. 

The  following  example  is 
from  Eutocius.  As  it  is  dim-    pvy  J//c/5 

cult  to  remember  the  value     —  

a  er  lm5,//3// 
of  all  the  Greek  characters,       '^pv  5"'2"'5"1"5' 

we  will  indicate  the  opera-       '  LTpvQ  3"l"5'9° 

tion  by  writing  1°,  2°,  3°,     j^q  2to3/,/4,/  9° 

etc.,  for  the  series  of  units 


GREEK    ARITHMETIC.  189 

1',  2',  3',  etc.,  for  the  series  of  tens;  1'',  2",  etc.,  for  the 
hundreds,  etc.,  and  denote  the  myriads  by  writing  m  as  an 
exponent. 

Division. — The  division  of  the  Greeks  was  still  more  intri- 
cate than  their  multiplication,  for  which  reason  it  seems  they 
generally  preferred  the  sexagesimal  division,  and  no  example 
is  left  at  length  by  any  of  those  writers  except  in  the  latter 
form ;  but  these  are  sufficient  to  throw  some  light  on  the  pro- 
cess they  followed  in  the  division  of  common  numbers,  and 
Delambre  has  accordingly  supposed  the  following  example : 

TA/ty-«0(  auxy  332",3"'3"2'90(l"/8"2'30 

P^Py      ~aumy  182    3  l,/,8'/2/3° 

pv.rnd 

Pfie.rjv 


ev£0 


150 
145 

0 

8 

3  2  9 
4 

4 

3 

1 
6 

9  2  9 
4  6 

5 
5 

4  6  9 
4  6  9 

This  example  will  be  found,  on  a  slight  inspection,  to  resem- 
ble our  compound  division,  or  that  sort  of  division  that  we 
must  necessarily  employ,  if  we  were  to  divide  feet,  inches 
and  parts  by  similar  denominations,  which,  together  with  the 
number  of  different  characters  that  they  made  use  of,  must 
have  rendered  this  rule  extremely  laborious ;  and  that  for  the 
extraction  of  the  square  root  was,  of  course,  equally  difficult, 
though  the  principle  was  the  same  as  ours,  except  in  the 
difference  of  the  notation.  It  appears,  however, that  they  fre- 
quently, instead  of  making  use  of  the  rule,  found  the  root  by 
successive  trials,  and  then  squared  it  in  order  to  prove  the  truth 
of  their  assumption. 

This  beautiful  system  was  vastly  superior  in  simplicity  and 
practical  utility  to  that  transmitted  to  and  retained  by  the 
Romans,  and  by  them  bequeathed  to  the  nations  of  modern 
Europe.  It  was,  at  least  when  it  had  reached  its  highest 
development    through  the  genius  of  Archimedes  and  Apollo 


140  THE    PHILOSOPHY    OF    ARITHMETIC. 

nius,  quite  well  fitted  for  an  instrument  of  calculation ;  and 
though  somewhat  cumbrous  in  its  structure,  was  capable  of 
performing  operations  of  very  considerable  difficulty  and  mag- 
nitude. 

It  will  be  seen,  however,  that  though  much  more  refined  and 
pliant  than  that  of  the  Romans,  the  notation  of  the  Greeks  is 
very  much  inferior  to  the  common  or  Hindoo  method  ;  and  one 
cannot  help  wondering  that  so  ingenious  and  philosophical  a 
people  failed  to  conceive  the  simple  idea  of  place  value,  and 
construct  a  system  of  notation  upon  it.  This  seems  all  the 
more  astonishing  when  we  remember  that  Archimedes  invented 
a  system  of  octates,  or  system  of  eights,  which  was  subse- 
quently improved  by  Apollonius,  by  making  the  periods  con- 
sist of  only  four  places,  and  dividing  all  numbers  into  orders 
of  myriads.  In  this  form,  as  Barlow  remarks,  it  seems  most 
astonishing  that  he  did  not  perceive  the  advantage  of  making 
the  periods  to  consist  of  a  less  number  of  characters;  for,  hav- 
ing given  a  local  character  to  his  periods  of  four,  it  was  only 
necessary  to  have  done  the  same  for  the  single  digits,  in  order 
to  have  arrived  at  the  system  in  present  use.  And  this  is 
the  more  singular,  as  the  use  of  the  cipher  was  not  unknown 
to  the  Greeks,  being  always  employed  in  their  sexagesimal 
operations  where  it  was  necessary ;  and  consequently  the  step 
between  this  improved  form  of  their  notation  and  that  of  the 
present  system  was  extremely  small,  although  the  advantages 
of  the  latter  when  compared  with  the  former  are  incalculably 
great.  It  seems  to  have  been  the  lot  of  the  metaphysical 
mind  of  the  Hindoos  to  make  this  "brilliant  invention  of  the 
decimal  scale,"  one  of  the  greatest  improvements  in  the  whole 
circle  of  the  sciences,  and  to  which  we  are  indebted  for  all  the 
remarkable  advances  in  modern  analysis. 


CHAPTER  VIII. 

ROMAN   ARITHMETIC. 

rpHE  arithmetic  of  the  Romans  was  quite  inferior  to  that  of 
JL  the  Greeks,  a  necessary  consequence  of  the  inferiority  of 
the  method  of  notation  adopted.  The  method  of  notation, 
though  usually  ascribed  to  the  Romans,  was  probably  invented 
by  the  Greeks,  and  communicated  by  them  to  the  Romans,  who 
in  turn  transmitted  it  to  their  successors  in  modern  Europe. 
It  no  doubt  originated  in  the  use  of  simple  strokes,  variously 
combined,  to  represent  numbers.  Subsequently  it  was  found 
convenient  to  represent  numbers  by  the  letters  of  the  alphabet, 
and  the  numerical  strokes  were  finally  displaced  by  such  alpha- 
betic characters  as  most  nearly  resembled  them. 

The  origin  of  the  Roman  characters  is  not  certainly  known; 
but  the  theory,  as  given  by  Leslie,  and  by  many  regarded  as 
correct,  is  interesting  and  plausible.  It  is  certain  that  the 
first  numerical  characters  consisted  simply  of  strokes  or  straight 
lines.  This  was  the  method  primarily  used  by  nearly  every 
nation  of  antiquity,  and  was  the  beginning  of  a  philosophical 
and  universal  system  alike  intelligible  to  all  nations.  Such 
characters  are  still  preserved  in  the  Roman  notation  with  very 
little  change,  and  were  probably  adopted  before  the  importation 
of  the  alphabet  itself,  by  the  Grecian  colonies  that  settled 
Italy  and  founded  the  Latin  commonwealth.  Assuming,  then, 
a  perpendicular  line  |  to  signify  one,  two  such  lines  |  |  to  signify 
tivo,  three  lines  1 1  |  to  signify  three,  and  so  on  up  to  ten,  and  we 
have  the  first  series  of  the  numerical  scale.     They  might  then 

(141) 


142  THE    PHILOSOPHY    OF    ARITHMETIC. 

be  supposed  to  throw  a  dash  across  the  last  stroke  or  unit,  to 
mark  the  completion  of  the  series;  and  thus,  a  cross,  X, 
would  come  to  signify  ten.  The  continued  repetition  of  this 
mark  would  denote  twenty,  thirty,  etc.,  until  they  reached  a 
hundred,  or  ten  tens,  which  completes  the  second  series,  and 
mignt  be  denoted  by  adding  another  dash  to  the  mark  for  ten, 
or  by  merely  connecting  three  strokes,  thus  C  The  repetition 
of  this  symbol  would,  in  like  manner,  indicate  the  successive 
hundreds,  the  tenth  of  which  would  be  marked  by  the  addition 
of  another  stroke,  so  that  four  combined  strokes,  M>  would 
express  a  thousand. 

Such  were  probably  the  symbols  originally  employed  in  the 
Roman  notation ;  in  process  of  time  it  would  be  perceived  that 
the  inconvenience  in  writing,  arising  from  so  many  repetitions 
of  the  same  character,  might  be  avoided  by  adopting  symbols 
for  the  intermediate  numbers;  and  it  was  seen  that  these 
might  be  furnished  by  the  division  of  the  symbols  already  in 
use.  Thus,  having  parted  in  the  middle  the  two  strokes,  X, 
either  the  under  half,  /\,  or  the  upper  half,  \J ,  was  employed 
to  signify  five,  or  the  half  of  ten.  Next,  for  fifty,  the  half  of 
a  hundred,  the  symbol  q  was  dirided  into  two  equal  parts,  [~ 
and  [_,  either  of  which  represented  fifty.  Again,  the  symbol 
for  thousand  having  come  to  assume  a  rounded  shape,  thus 
p) ,  or  thus  CD,  the  half  of  this,  either  CI,  or  13,  was  taken  to 
represent  the  half  of  one  thousand  or  five  hundred.  The 
symbol  Q,  to  represent  a  hundred,  would,  in  process  of  time, 
being  frequently  made,  have  its  corners  rounded  and  attain 
the  form  C.  Lastly,  noticing  that  these  characters  closely 
resemble  some  of  the  letters  of  the  alphabet,  it  was  agreed  to 
employ  those  letters  as  the  symbols  of  the  numbers  mentioned. 

The  notation  of  numbers  by  combined  strokes,  was  evi- 
dently founded  in  nature,  and  may  be  regarded  as  the  begin- 
ning of  a  philosophical  language  of  arithmetic.  That  this 
was  the  foundation  of  the  Roman  system. is  confirmed  from  the 
analogous  practice  of  other  nations.     It  is  quite  clear  that  the 


ROMAN   ARITHMETIC.  145 

Egyptians  and  Chinese  must  have  followed  the  same  method. 
The  inscriptions  on  the  ancient  obelisks  present  a  few  numerals 
which  are  easily  distinguished.  The  substitution  of  capital 
letters  for  the  combined  strokes  which  they  chanced  most  to 
resemble,  though  it  gave  uniformity  to  the  system  of  notation, 
prevented  any  farther  improvements  of  the  system.  The  only 
simplification  which  the  Romans  appear  to  have  introduced, 
was  to  diminish  the  repetition  of  letters  by  reckoning  in  some 
cases  backwards,  as  in  IV,  which  was  originally  represented 
by  four  strokes,  and  IX,  which  was  probably  at  first  written 

Villi. 

Their  method  of  representing  large  numbers  was  a  little 
different  from  that  now  used,  as  may  be  seen  by  the  following 
examples : 

Dor  D       MorCID       133       CCI33       1333       CCCI333 
500  1000  5000      10,000     50,000     100,000. 

In  illustration,  it  is  interesting  to  notice  that  Cicero  in  his 
fifth  oration  against  Yerres  expresses  3600  by  C13  C13  CI3  I3C 
The  Romans  often  contracted  or  modified  the  forms  of  their 
numerals,  especially  in  carving  inscriptions  upon  stones,  in 
which  case  the  abbreviated  letters  were  called  lapidary  char- 
acters. 

The  marks  for  any  number  could  also  be  augmented  in  power 
one  thousand  times,  either  by  enclosing  them  with  two^hooks  or 
C's,  or  by  drawing  a  line  over  them.  Thus,  CX3,  or  X  denoted 
10,000,  and  CLVIM  given  by  Pliny,  means  156,000,000. 
Sometimes  a  letter  was  placed  over  another  to  indicate  their 
product ;  thus,  ^  would  express  500,000.  The  multiplier  was 
also  sometimes  written  like  an  exponent,  thus  IIP  was  used 
to  express  three  hundred.  In  expressing  very  large  numbers, 
points  were  sometimes  interposed:  thus,  Pliny  writes  XVI. 
XX.DCCCXXIX  for  1,620,829.  It  may  be  remarked  that  if  this 
practice  had  become  more  general  it  would  probably  have 
effected  a  material  improvement  of  the  system. 


144  THE    PHILOSOPHY   OF    ARITHMETIC. 

In  the  latter  ages  of  the  Roman  Empire,  the  small  letters 
of  the  alphabet  seem  to  have  been  used  in  imitation  of  the 
numeral  system  of  the  Greeks.  The  letters  a,  b,  c,  d,  e,  f, 
g,  h,  and  i  represented  the  nine  digits  1,  2,  3,  4,  5,  6,  7,  8,  and  9  ; 
the  next  series  k,  1,  m,  n,  o,  p,  q,  r,  and  s  expressed  10,  20,  30, 
40,  50,  60,  TO,  80,  and  90;  and  the  remaining  letters  t,  u,  x, 
y,  and  z  denoted  100,  200,  300,  400,  and  500.  To  represent 
the  rest  of  the  hundreds  it  was  necessary  to  employ  capitals 
or  other  characters,  and  600,  700,  800,  and  900  were  repre- 
sented by  I,  Y,  hi  and  hu.  But  this  mode  of  notation  never 
obtained  any  degree  of  currency,  being  mostly  confined  to 
those  foreign  adventurers  from  Greece,  Egypt  or  Chaldea,  who, 
pretending  to  skill  in  judicial  astrology,  were  enabled  to  prey 
on  the  credulity  of  the  wealthy  Romans. 

In  modern  Europe  the  Roman  numerals  were  supplied  by 
Saxon  characters.     Thus,  in  the  accounts  of  the  Scottish  Ex- 
chequer for  the  year  1331,  the  sum  of  £6896   5s.  5d.  stated  as 
paid  to  the  King  of  England  is  thus  marked: 
D        c        xx 
vj.    viij.      iiij.     xvj.     rj.     v.      s.     v.      d. 

The  Roman  system,  as  now  used,  employs  seven  characters, 
of  which  I  represents  one,  Y  five,  X  ten,  L  fifty,  C  one  hun- 
dred, D  five  hundred,  M  one  thousand.  To  express  other 
numbers  these  characters  are  combined  according  to  the  fol- 
lowing principles: — 

1.  Every  time  a  letter  is  repeated  its  value  is  repeated. 

2.  When  a  letter  is  placed  after  one  of  greater  value,  the 
sum  of  their  values  is  the  number  expressed. 

3.  When  a  letter  is  placed  before  one  of  a  greater  value,  the 
difference  of  their  values  is  the  number  expressed. 

4.  When  a  letter  stands  between  two  letters  of  a  greater 
value,  it  is  combined  with  the  one  following  it. 

5.  A  letter  is  placed  before  one  of  its  own  order  only,  or  the 
unit  of  the  next  higher  order. 

6.  A  dash  over  a  letter  increases  its  value  a  thousand    fold. 


ROMAN    ARITHMETIC.  l_f> 

In  accordance  with  the  fifth  principle  it  would  be  incorrect 
to  write  VC  for  ninety-five,  or  IC  for  ninety-nine.  It  is  also 
to  be  noticed  that  the  letter  V  is  never  used  before  a  letter  of 
greater  value,  since  the  only  case  in  which  it  could  be  thus 
used  according  to  the  fifth  principle  is  before  X,  giving  VX 
for  five,  which  is  more  concisely  expressed  by  V  itself. 

In  expressing  numbers  by  the  Roman  method  we  always 
write  the  different  orders  of  units  successively,  beginning  with 
the  higher  orders.  Thus,  in  expressing  four  hundred  and 
ninety-nine,  we  would  not  write  ID,  though  this,  by  principle 
second,  would  be  the  difference  of  one  and  five  hundred,  but 
we  first  write  CCCC  for  four  hundred,  then  XC  for  ninety , 
and  then  IX  for  nine,  giving  CCCCXCIX. 

It  may  be  interesting  to  notice,  however,  that  though  the 
Roman  method  was  not  employed  in  numerical  calculations,  it 
might  have  been  so  employed  by  slightly  modifying  the  usual 
mode  of  notation.  Thus,  by  not  using  the  third  principle,  but 
writing  IIII  for  IV,  and  YIIII  for  IX,  or  by  using  some 
mark  to  show  that  the  letters  written  according  to  that  prin- 
ciple are  taken  together,  as  XXIV,  we  can  perform  the  four 
fundamental  operations  without  much  inconvenience.  To  illus- 
trate, we  give  a  problem  in  multiplication,  with  its  explanation. 

Explanation. — VIII    multiplied 

.       _£_  ,      T_T    v  ,  .    ,.     ,  OPERATION. 

by  VII  equals  LVI,  X  multiplied 

by  VII  equals  LXX,  L  multiplied  XXXVII 

by  VII  equals  CCCL  ;    III  multi- 

plied  by  X  equals  XXX,  X  multi-  DCLX_£_f LVI 

plied  by  X  equals  C,  L  multiplied  DCLXXX 

by  X  equals  DCL ;  multiplying  by        DCLXXX 

X  a   second  and   third   time,  and        \f\m XVI 

taking  the  sum  of  the  four  partial 

products,  we  have  MMDXVI,  or  two  thousand  five  hundred 
and  sixteen.     This  result  may  be  obtained  by  multiplying  by 
VII  and  XXX;  or  by  II,  V,  X,  and  XX,  etc.     The  multiple 
cand  also  may  be  variously  separated  in  the  multiplication. 
10 


146  THE    PHILOSOPHY    OF    ARITHMETIC. 

It  is  clear,  however,  that  this  operation  would  be  very  com- 
plicated with  large  numbers,  so  much  so,  indeed,  as 
to  be  unfitted  for  general  use,  and  it  is  believed  that  it  was  not 
used  in  performing  numerical  calculations.  These  calculations 
were  performed  by  means  of  counters,  or  other  palpable  em- 
blems. The  instrument  generally  used  was  called  the  Abacus. 
Leslie  says  that  "the  system  of  characters  among  the  Romans 
was  so  complex  and  unmanageable  as  to  reduce  them  to  the 
necessity  in  all  cases  of  employing  the  Abacus." 

The  Abacus  appears  to  have  continued  in  use  among  the 
people  of  Europe  until  quite  a  recent  period.  The  counters  or 
pebbles  were,  from  a  corruption  of  the  word  algorithm,  called 
in  England  augrim,  or  awgrym,  stones.  Thus,  in  Chaucer's 
description  of  the  chamber  of  Clerk  Nicholas,  he  says: 

"  His  almageste  and  bokes  grete  and  smale, 
Hisastrelabre,  longing  for  his  art, 
His  augrim  stones  Layen  faire  apart 
On  shelves  couched  at  his  beddes  head." 

Indeed,  the  modern  method  of  arithmetic  was  not  known  in 
England  until  about  the  middle  of  the  sixteenth  century ;  and 
the  common  people,  imitating  the  clerks  of  former  times,  were 
still  accustomed  to  reckon  by  the  help  of  the  awgrym  stones. 
Thus,  in  Shakespeare's  comedy  of  the  Winter's  Tale,  written 
at  the  beginning  of  the  seventeenth  century,  a  clown,  staggered 
at  a  very  simple  multiplication,  exclaims  that  he  must  try  it 
with  counters. 

Clo.  Let  me  see  ;  Every  'leven  wether  —  tods  ;  every  tod  yields 
—  pound  and  odd  shilling;  fifteen  hundred  shorn, — What  comes 
the  wool  to?    .     .     .     I  cannot  do't  without  counters. 

The  Roman  method  is  now  chiefly  used  to  denote  the  vol- 
umes, chapters,  sections  and  lessons  of  books,  the  pages  of  pre- 
faces and  introductions,  to  express  dates,  to  mark  the  hours  on 
clock  and  watch  faces,  and  in  other  places  for  the  sake  of  prom- 
inence and  distinction. 


CHAPTER  IX 

PALPABLE    ARITHMETIC. 

fpiIE  earliest  methods  of  representing  numbers  in  arithmetical 
-L  calculation  were  by  means  of  counters  and  other  palpable 
emblems.  The  objects  most  generally  used  among  all  primitive 
nations  were  little  stones  or  pebbles,  from  which  we  derive  our 
word  calculation.  Beginning  with  pebbles  or  some  such  sim- 
ple objects,  as  they  advanced  in  civilization  these  were  found  to 
be  insufficient  for  their  purposes,  and  they  invented  instruments 
to  represent  numbers,  by  means  of  which  they  were  enabled 
to  calculate  with  great  rapidity  and  correctness.  The  Japan- 
ese and  Chinese  at  the  present  day,  with  their  arithmetical 
instruments,  can  add,  subtract,  multiply  and  divide  as  rapidly 
and  correctly  as  we  can  with  the  Arabic  system  of  notation. 
So  extensively  was  this  method  used  by  the  early  nations  before 
the  method  of  calculating  by  figures  was  adopted,  that  Leslie, 
in  his  treatise  on  arithmetic,  gives  it  a  distinct  and  detailed 
explanation  under  the  head  of  Palpable  Arithmetic.  The  sub- 
ject is  so  full  of  interest,  both  for  its  own  ingenuity  and  its 
relation  to  our  present  system,  that  I  think  it  proper  to  devote 
a  chapter  to  it,  and  finding  a  clearer  statement  of  it  in  Leslie 
and  Peacock  than  I  could  hope  to  give  myself,  I  have  tran- 
scribed their  description,  sometimes  word  for  word. 

The  early  Egyptians  performed  their  computations  mainly 
by  the  help  of  pebbles,  and  so  did  the  early  Greeks  and 
Romans.  In  the  schools  of  ancient  Greece,  the  boys  acquired 
the  elements  of  knowledge  by  working  on  the  ABAX,  a  smooth 

(147) 


148  THE   PHILOSOPHY    OF   ARITHMETIC. 

board  with  narrow  rim,  so  named  evidently  from  the  combina- 
tion of  the  first  three  letters  of  their  alphabet,  and  resembling 
the  tablet  on  which  children  were  formerly  accustomed  to  begin 
to  learn  the  art  of  reading.  Pupils  were  taught  to  calculate  by 
forming  progressive  rows  of  counters,  which  consisted  of  round 
bits  of  bone  or  ivory,  or  even  silver  coins,  according  to  the 
wealth  or  fancy  of  the  individual.  The  same  board,  strewed 
with  fine  green  sand,  a  color  soft  and  agreeable  to  the  eye, 
served  equally  for  teaching  the  rudiments  of  writing  and  the 
principles  of  geometry. 

The  ancient  writers  make  frequent  allusions  to  these  calculat- 
ing boards.  Solon,  the  great  Athenian  statesman,  used  to 
compare  the  passive  ministers  of  kings  to  the  counters  or 
pebbles  of  arithmeticians  which,  according  to  the  place  they 
hold,  are  sometimes  most  important,  and  sometimes  utterly 
insignificant.  The  Grecian  orators,  in  speaking  of  balanced 
accounts,  picture  the  settlements  by  saying  that  the  pebbles 
were  cleared  away  and  none  left.  It  thus  appears  that  the 
ancients,  in  keeping  their  accounts,  did  not  arrange  the  debits 
and  credits  separately,  but  set  down  pebbles  for  the  former,  and 
took  up  pebbles  for  the  latter.  As  soon  as  the  board  became 
cleared,  the  opposite  claims  were  exactly  balanced.  It  may 
be  observed  that  the  common  phrase  to  clear  one's  scores  or 
accounts,  meaning  to  settle  or  adjust  them,  still  preserved  in  the 
popular  language  of  Europe,  was  suggested  by  the  same  prac- 
tice of  reckoning  with  counters,  which  prevailed,  indeed,  until 
a  comparatively  late  period. 

The  Romans  borrowed  their  Abacus  from  the  Greeks,  and 
seem  never  to  have  aspired  higher  in  the  pursuit  of  numerical 
science.  To  each  pebble  or  counter  required  for  the  board 
they  gave  the  name  of  calculus,  meaning  a  small  white  stone, 
and  applied  the  verb  calculare  to  express  the  operation  of  com- 
bining or  separating  such  pebbles  or  counters.  The  use  of 
the  Abacus,  called  also  the  Mensa  Pythagorica,  formed  an 
essential  part  of  the  education  of  every  noble  youth.    A  small 


PALPABLE    ARITHMETIC.  149 

box  or  coffer,  called  a  Loculus,  having  compartments  for  hold- 
ing the  calculi,  or  counters,  was  considered  as  a  necessary 
appendage.  Instead  of  carrying  a  slate  and  satchel  to  school, 
the  Roman  boy  was  accustomed  to  trudge  to  school  loaded 
with  those  ruder  implements, — his  arithmetical  board  and  his 
box  of  counters. 

In  the  progress  of  luxury  and  refinement,  dice  made  of  ivory, 
called  tali,  were  used  instead  of  pebbles,  and  small  silver 
coins  came  to  supply  the  place  of  counters.  Under  the  Em- 
perors, every  patrician  living  in  a  spacious  mansion  and 
indulging  in  all  the  pomp  and  splendor  of  Eastern  princes, 
generally  entertained,  for  various  functions,  a  numerous  train 
of  foreign  slaves  or  freedmen  in  his  palace.  Of  these,  the 
librarius,  or  miniculator,  was  employed  in  teaching  the 
children  their  letters,  the  notarius  registered  expenses,  the 
rationarius  adjusted  and  settled  accounts,  and  the  tabularius 
or  calculator,  working  with  his  counters  and  board,  performed 
what  computations  might  be  required. 

To  facilitate  the  working  by  counters,  the  construction  of  the 
Abacus  was  afterwards  improved.  Instead  of  the  perpendic- 
ular lines,  or  bars,  the  board  had  its  surface  divided  by  sets  of 
parallel  grooves,  by  stretched  wires,  or  even  by  successive 
rows  of  holes.  It  was  easy  to  move  small  counters  in  the 
grooves,  to  slide  perforated  beads  along  the  wires,  or  to  stick 
large  knobs  or  round-headed  nails  in  the  different  holes.  To 
diminish  the  number  of  marks  required,  every  column  was 
surmounted  by  a  shorter  one,  wherein  each  counter  had  the 
same  value  as  five  of  the  ordinary  kind.  The  Abacus,  instead 
of  wood,  was  often,  for  the  sake  of  convenience  and  durabil- 
ity, made  of  metal,  frequently  brass,  and  sometimes  silver. 
Two  varieties  of  this  instrument  seem  to  have  been  used  by 
the  Romans.  Both  of  them  are  delineated  from  antique 
monuments — the  first  kind  by  Ursinus,  and  the  second  by 
Marcus  Yelserus.  In  the  former,  the  numbers  are  represented 
by  flattish  perforated  beads,  ranged  on  parallel  wires;  and  in 


150  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  latter,  they  are  signified  by  small  round  counters,  moving 
in  parallel  grooves.  These  instruments  contain  each  seven 
capital  divisions,  expressing  in  regular  order  units,  tens,  hun- 
dreds, thousands,  ten  thousands,  hundred  thousands,  and 
millions,  and  as  many  shorter  divisions,  of  five  times  the  rela- 
tive value  of  the  larger  ones.  With  four  beads,  on  each  of 
the  long  grooves  or  wires,  and  one  on  each  corresponding  short 
one,  it  is  evident  that  any  number  could  be  expressed  up 
to  ten  millions.  The  Roman  Abacus  also  contained  grooves 
to  mark  ounces,  half-ounces,  quarter-ounces,  and  thirds  of  an 
ounce. 

The  Ilomans  likewise  applied  the  same  word  Abacus  to  an 
article  of  furniture  resembling  in  shape  the  arithmetical  board, 
but  often  highly  ornamented,  which  was  destined  for  the 
amusement  of  the  opulent.  It  was  used  in  a  game  apparently 
similar  to  that  of  chess,  in  which  the  infamous  and  abandoned 
Nero  took  particular  delight,  driving  over  the  surface  of  the 
Abacus  with  a  beautiful  ivory  quadriga  or  chariot. 

The  Chinese  have,  from  the  remotest  ages,  used  in  all  their 
computations,  an  instrument  similar  in  shape  and  construction 
to  the  Roman  Abacus,  but  more  complete  and  uniform.  It 
is  admirably  adapted  to  the  decimal  system  of  weights,  meas- 
ures, and  coins,  which  prevails  throughout  the  empire.  The 
whole  range  includes  ten  bars,  and  the  calculator  may  begin 
at  any  one  and  reckon  upwards  or  downwards  with  equal 
facility,  treating  fractions  exactly  like  integers — an  advantage 
of  the  utmost  consequence  in  practice.  Accordingly  these 
arithmetical  machines,  of  various  sizes,  have  been  adopted  by 
all  ranks,  from  the  man  of  letters  to  the  humblest  shopkeeper, 
and  are  constantly  used  in  all  the  bazaars  and  booths  of  Can- 
ton and  other  cities,  being  handled,  it  is  said,  by  the  native 
traders  with  a  rapidity  and  address  quite  astonishing. 

Among  the  various  nations  which  regained  their  independ- 
ence by  the  fall  of  the  Roman  Empire,  it  was  found  convenient 
in  all  transactions  where  monev  was  concerned,  to  follow  the 


PALPABLE    ARITHMETIC.  151 

procedure  of  the  Abacus,  in  representing  numbers  by  counters 
placed  in  parallel  rows.  During  the  Middle  Ages,  it  became 
the  usual  practice  over  Europe  for  merchants,  auditors  of 
accounts,  or  judges  appointed  to  decide  in  matters  of  revenue, 
to  appear  on  a  covered  bank  or  bench,  so  called  from  an  old 
Saxon  or  Franconian  word  signifying  a  seat.  The  term 
scaccarium,  a  Latinized  Oriental  word,  from  which  was 
derived  the  French  and  then  the  English  name  for  the 
Exchequer,  anciently  indicated  merely  a  chess-board,  being 
formed  from  scaccum,  one  of  the  pieces  in  that  game. 

The  Court  of  Exchequer,  which  takes  cognizance  of  all 
questions  of  revenue,  was  introduced  into  England  by  the 
Norman  Conquest.  Fitz-Nigel,  in  a  dialogue  on  the  subject, 
written  about  the  middle  of  the  twelfth  century,  says  that  the 
scaccarium  was  a  quadrangular  table  about  ten  feet  long  and 
five  feet  broad,  with  a  ledge  or  border  about  four  inches  high, 
to  prevent  anything  from  rolling  over,  and  was  surrounded  on 
all  sides  by  seats  for  the  judges,  the  tellers,  and  other  officers. 
It  was  covered  every  year,  after  the  term  of  Easter,  with  fresh 
black  cloth,  divided  by  perpendicular  white  lines  or  distinc- 
tures,  at  intervals  of  about  a  foot  or  a  palm,  and  again  parted  by 
similar  transverse  lines.  In  reckoning  accounts,  they  pro- 
ceeded according  to  the  rules  of  arithmetic,  using  small  coins 
for  counters.  The  lowest  bar  exhibited  pence,  the  one  above 
it  shillings,  the  next  pounds,  and  the  higher  bars  denoted  suc- 
cessively lens,  twenties,  hundreds,  thousands,  and  ten  thou- 
sands  of  pounds;  though,  in  those  early  times  of  penury  and 
severe  economy,  it  very  seldom  happened  that  so  large  a  sum 
as  the  last  ever  came  to  be  reckoned.  The  teller  sat  about  the 
middle  of  the  table ;  on  his  right  hand,  eleven  pennies  were 
heaped  on  the  first  bar,  and  nineteen  shillings  on  the  second, 
while  a  quantity  of  pounds  was  collected  opposite  to  him,  on 
the  third  bar.  For  the  sake  of  expedition  he  might  employ  a 
different  mark  to  represent  half  the  value  of  any  bar,  a  silver 
penny  for  ten  shillings,  and  a  gold  penny  for  ten  pounds. 


162  THE    PHILOSOPHY    OF    ARITHMETIC. 

In  early  times,  a  checkered  board,  the  emblem  of  calculation, 
was  hung  out,  to  indicate  an  office  for  changing  money.  It 
was  afterwards  adopted  as  the  sign  of  an  inn  or  hostelry, 
where  victuals  were  sold,  or  strangers  lodged  and  entertained. 
It  is  said  that  traces  of  this  ancient  practice  may  be  found  even 
at  the  present  day. 

The  use  of  the  smaller  Abacus  in  assisting  numerical  com- 
putation was  not  unknown  during  the  Middle  Ages.  In 
England,  however,  it  appears  to  have  scarcely  entered  into 
actual  practice,  being  mostly  confined  to  those  few  individuals 
who,  in  such  a  benighted  period,  passed  for  men  of  science 
and  learning.  The  calculator  was  styled,  in  correct  Latin, 
abacista;  but  in  Italian,  abbachista,  or  abbachiere.  The 
Arabians,  having  adopted  an  improved  species  of  numeration, 
to  which  they  gave  the  barbarous  name  of  algarismus  or  algo- 
rithms, from  their  definite  article  al,  and  the  Greek  word  for 
number,  this  compound  term  was  adopted  by  the  Christians 
of  the  West,  in  admiration  of  their  superior  skill,  to  signify 
calculation  in  general,  long  before  the  peculiar  method  of  per- 
forming it  had  become  known  and  practiced  among  them. 
The  term  algarism  was  converted  in  English  into  augrim 
or  awgrym,  and  applied  even  to  the  pebbles  or  counters  used 
in  ordinary  calculation.  The  same  word,  algorithm,  is  now 
applied  by  mathematicians  to  express  any  peculiar  sort  of 
notation. 

The  Abacus  had  been  adopted  merely  as  an  instrument 
for  facilitating  the  process  of  computation.  It  became 
necessary,  however,  to  adopt  some  simpler  and  more  conveni 
ent  method  of  expressing  numbers.  A  very  ancient  practice 
consisted  in  employing  the  various  articulations  and  disposi- 
tions of  the  fingers  and  the  hands,  to  denote  the  numerical 
3eries.  On  this  narrow  basis,  the  Romans  framed  a  system  of 
considerable  extent.  By  the  inflexion  of  the  various  fingers 
of  the  left  hand,  they  proceeded  as  far  as  ten,  and  by  combin- 
ing these  with  some  other  given  inflexions,  as  changes  in  the 


PALPABLE    ARITHMETIC.  153 

position  of  the  thumb,  they  could  advance  to  a  hundred ;  and 
using  the  right  hand  in  a  similar  manner,  they  proceeded  as 
far  as  a  thousand  and  ten  thousand.  This  is  as  far  as  the 
system  appears  to  have  been  carried  by  the  ancients ;  but  the 
venerable  Bede,  by  referring  these  signs  to  the  various  parts 
of  the  body,  as  the  head,  the  throat,  the  side  of  the  chest,  the 
stomach,  the  waist,  the  thigh,  etc.,  has  shown  how  they  could 
be  again  multiplied  a  hundred  times,  and  raised  to  the  extent 
of  a  million.  In  this  numerical  play,  the  Romans  especially 
had  acquired  great  dexterity.  Many  allusions  to  the  practice 
are  made  by  their  poets  and  orators,  and  without  some  knowl- 
edge of  the  principle  adopted,  many  passages  of  the  classics 
would  lose  their  whole  force. 

A  species  of  digital  arithmetic  seems  to  have  existed  among 
nearly  all  the  Eastern  nations.  The  Chinese  have  a  system  of 
indigitation  by  which  they  can  express  on  one  hand  all  num- 
bers less  than  100,000  The  thumb  nail  of  the  right  hand 
touches  each  joint  of  the  little  finger,  passing  first  up  the 
external  side,  then  down  the  middle,  and  afterwards  up  the 
other  side  of  it,  in  order  to  express  the  nine  digits;  the  tens 
are  denoted  in  the  same  way  on  the  second  finger ;  the  hun- 
dreds on  the  third;  the  thousands  on  the  fourth;  the  tens  of 
thousands  on  the  thumb.  It  would  be  only  necessary  to  pro- 
ceed to  the  right  hand  in  order  to  be  able  to  extend  this  system 
of  numeration  much  further  than  could  be  required  for  any 
ordinary  purposes.  The  Bengalese  count  as  far  as  15  bv 
touching  in  succession  the  joints  of  the  fingers  ;  and  merchants 
in  concluding  bargains,  the  particulars  of  which  they  wish 
to  conceal  from  the  by  standers,  put  their  hands  beneath  a 
cloth  and  signify  the  prices  they  offer  or  take  by  the  contact 
of  the  fingers.  The  same  custom  is  prevalent  also  in  Barbary 
and  Arabia,  where  they  conceal  their  hands  beneath  the  folds 
of  their  cloaks,  and  possess  methods  which  are  probably  pecu- 
liar and  national,  of  conveying  the  expression  of  numbers  to 
each  other. 


154  THE    PHILOSOPHY    OF    ARITHMETIC. 

Juvenal  states  it  as  a  peculiar  felicity  of  Nestor  tbat  be 
counted  the  years  of  bis  age  on  bis  right  band.  Tbe  image  of 
Janus  was  represented,  according  to  Pliny,  witb  tbe  fingers 
so  placed  as  to  represent  365,  tbe  number  of  days  in  tbe  year. 
Some  autbors  bave  supposed  tbat  Solomon  in  tbe  passage, 
"Length  of  days  is  in  her  right  hand,  and  in  her  left  hand 
riches  and  honor,"  referred  to  this  practice.  The  common 
phrases,  ad  digitos  redire,  in  digitos  mittere,  have  the  same 
meaning  as  computare,  and  distinctly  refer  to  digital  numera- 
tion ;  and  tbe  phrase  micare  digitis,  of  frequent  occurrence, 
alludes  to  a  game  extremely  popular  among  the  Romans, 
and  which  was  probably  tbe  same  as  the  morra  of  modern 
Italy.  This  noisy  game  is  played  by  two  persons,  who  stretch 
out  a  number  of  their  fingers  at  the  same  moment,  and  instantly 
call  out  a  number;  and  be  is  the  winner  who  expresses  the 
sum  of  the  number  of  fingers  thrown  out.  The  same  game 
is  found  amongst  the  Sicilians,  Spaniards,  Moors,  and  Persians, 
and  under  tbe  name  tsoimoi,  is  practiced  also  in  China. 

These  signs  were  merely  fugitive,  and  it  became  necessary 
to  adopt  other  marks  of  a  permanent  nature  for  the  purpose  of 
recording  numbers.  But  of  all  the  contrivances  adopted  with 
this  view,  the  rudest  undoubtedly  is  the  method  of  registering 
by  tallies,  introduced  into  England  along  with  the  Court  of 
Exchequer,  as  another  badge  of  tbe  Norman  Conquest.  These 
consist  of  straight,  well-seasoned  sticks  of  hazel  or  willow,  so 
called  from  the  French  verb  tailler,  to  cut,  because  they  are 
squared  at  each  end.  The  sum  of  money  was  marked  on  the 
side  with  notches,  by  the  cutter  of  tallies,  and  likewise  inscribed 
on  both  sides  in  Roman  characters,  by  the  writer  of  the  tallies. 
The  smallest  notch  signified  a  penny,  a  larger  one  a  shilling, 
and  one  still  larger  a  pound;  but  other  notches,  increasing  suc- 
cessively in  breadth,  were  made  to  denote  ten,  a  hundred,  and 
a  thousand.  The  stick  wras  then  cleft  through  the  middle  by 
the  deputy-chamberlains,  with  a  knife  and  mallet,  the  one  por- 
tion being  called  a  tally,  or  sometimes  the  scachia,  stipes,  or 


PALPABLE    ARITHMETIC.  155 

Icancia,  and   the    other   portion   named   the    counter-tally  or 
folium. 

This  strange  custom  might  seem  the  practice  of  untutored 
Indians,  and  can  be  compared  only  to  the  rude  simplicity  of 
the  ancient  Romans,  who  kept  their  diary  by  means  of  lapilli 
or  small  pebbles,  casting  a  white  pebble  into  the  urn  on  fortu- 
nate days,  and  dropping  a  black  one  when  matters  looked 
unprosperous ;  and  who  sent,  at  the  close  of  each  year,  the 
Praetor  Maximus,  with  great  solemnity,  to  drive  a  nail  in  the 
door  of  the  right  side  of  the  temple  of  Jupiter,  next  to  that  of 
Minerva,  the  patron  of  learning  and  inventor  of  numbers. 

The  use  of  counters  was  general  throughout  Europe  as  late 
as  the  end  of  the  15th  century:  about  that  period  they  were  no 
longer  used  in  Italy  and  Spain,  where  the  early  introduction 
of  the  Arabic  figures  and  the  number  of  treatises  on  the  use  of 
these  figures  had  rendered  them  unnecessary.  Recorde,  in  his 
Ground  of  Arts,  prefaces  his  second  dialogue,  entitled  "  The 
Accounting  by  Counters,"  by  observing,  "Now  that  you  have 
learned  Arithmetic  with  the  pen,  you  shall  see  the  same  art  in 
counters,  which  feat  doth  not  onely  serve  for  them  that  cannot 
write  and  read,  but  also  for  them  that  can  do  both,  but  have 
not  at  the  same  time  their  pen  or  tables  with  them." 

We  shall  now  proceed  to  give  some  account  of  the  method 
of  performing  operations  by  this  palpable  or 
calcular  arithmetic.  They  commenced  by 
drawing  seven  lines  with  a  piece  of  chalk, 
on  a  table,  board,  or  slate,  or  by  a  pen  on 
paper,  as  in  the  margin ;  the  counters,  which 
were  usually  of  brass,  on  the  lowest  line 
represented  units,  on  the  next  tens,  and  so 
on  as  far  as  millions  on  the  uppermost  line; 
a  counter  placed  between  two  lines  repre- 


■#— •- 


sented  five  counters  on  the  line  next  below 

it;  thus,  the  number  represented  in  the 
margin  is  3629638,  and  the  number  of  lines 
may  evidently  be  increased  so  as  to  represent  any  number. 


156 


THE    PHILOSOPHY    OF    ARITHMETIC. 


To  add  two  numbers,  such  as  788  and  383,  we  divide  the 
lines  as  in  the  margin,  so  as  to  form  three  columns,  writing 

the  first  number  in  the  first , , — •- 

column,  numbering  from  the  • 
left,  the  second  in  the  second,  ~"^— ' 
and  the  result  in  the  third 
column.  The  sum  of  the 
counters  on  the  lowest  line 
in  the  first  two  columns  is  6 ;  we  therefore  place  oue  on  that 
line  in  the  third  column,  and  carry  one  to  the  space  above 
which,  added  to  the  one  already  there,  makes  one  on  the  second 
line ;  adding  this  counter  to  the  six  already  there,  we  have 
7,  and  therefore  place  2  on  the  line  and  carry  one  to  the  space 
above ;  adding  the  counters  on  that  space,  we  find  there  are  3, 
hence  we  leave  one  in  the  space  and  carry  one  to  the  next  line, 
in  which  the  sum  of  the  counters  is  six ;  we  leave  one  on  the 
line  and  carry  one  to  the  space  above,  and  adding  to  the 
counter  already  there  we  have  two  counters,  hence  we  leave 
no  counter  there,  but  place  one  on  the  fourth  line ;  the  sum 
thus  obtained  will  be  1171. 

The  principle  of  this  operation  is  extremely  simple,  and  the 
process  could,  with  a  little  practice,  be  performed  with  much 
rapidity.  In  practice,  the  last  column  would  not  be  used,  as 
the  counters  on  each  line  would  be  removed  as  the  addition 
proceeded,  and  replaced  by  those  which  denoted  their  sum. 

We  will  illustrate  the  method  of  subtraction  by  taking  682 
from  1375.  The  two  count- 
ers on  the  first  line  have 
none  to  correspond  from 
which  theycan  be  subtracted ; 
we  therefore  bring  down 
the  counter  from  the  space 
above  and  replace  it  by  5  counters  on  the  line  ;  we  shall  then 
have  3  counters  left  on  the  line  and  none  on  the  space  ; 
we  then  bring  down  1  counter  from  the  second  space,  leaving 


-# . . 

•  •  • 

. L-0_# L#_#_# 


PALPABLE    ARITHMETIC. 


a  remainder  of  4  counters  on  the  line ;  then  bring  down  1 
counter  from  the  third  line  to  the  second  space,  and  we  have  1 
counter  left ;  and  so  we  proceed  until  the  subtraction  is  com- 
plete, and  we  shall  have  as  a  remainder  693.  Recorde  writes 
the  smaller  number  in  the  first  column,  and  commences  sub- 
tracting at  the  upper  line. 

To  illustrate  the  process  of  multiplication,  let  us  find  the 
product  of  2457  by  43.  We  express  the  multiplicand  in  the 
first  column  and  the 
multiplier  in  the 
second ;  multiply 
first  by  3,  and 
place  the  product 
in  the  third  column 
and  the  product  by 
4  in  the'  fourth; 
add  the  numbers  in 
these  two  columns, 
and  the  sum  is  the  product  required. 

Division  may  be  illustrated  by  dividing  12832  by  608. 
Since  six  hundreds  is  contained  in  12  thousands  2  tens  times, 
we  place  two 
counters  on  the 
second  line  of  the 
quotient;  multiply- 
ing 6  hundreds  by 
2  tens  and  subtract- 
ing, we  have  no  re- 
mainder; multiply- 
ing 8  by  2  tens,  we 
have  16  tens;  but  since  16  tens  equal  1  hundred  and  6  tens, 
we  take  off  1  from  the/3  in  the  third  or  hundreds  line,  leaviug 
2  remaining;  then  take  off  1  of  those  2  and  replace  it  by  2  in 
the  second  space,  and  then  take  1  from  the  second  space  and  1 
from   the   second  line;   then  transfer  the  remaining  counters 


• 

~»— 

• 

• 

• 

• 

© 

• 

• 

• 

• 
_#-c 

-•-#-• — 

-« 

-•— 

_# 

• 

• 

• 

• 

• 

-•H» 

• 

-• 

-♦-♦— J 

■»M  • 

158  THE    PHILOSOPHY    OF    ARITHMETIC. 

to  the  column  of  the  first  remainder,  and  we  have  as  a  re- 
mainder 672.  The  operation  is  repeated,  placing  the  quotient 
1  on  the  lowest  line  of  the  quotient  column ;  and  in  this  case 
we  merely  subtract  the  divisor  from  the  first  remainder,  obtain- 
ing 64  for  the  last  remainder,  and  21  for  the  quotient.  This 
process  may  evidently  be  repeated  to  any  extent ;  but  in  prac- 
tice it  was  much  simplified  by  removing  the  counters  of  the 
dividend  to  form  the  first  remainder,  and  so  on  until  the  opera- 
tion was  complete. 

Recorde  mentions  two  different  ways  of  representing  sums 
of  money  by  means  of  counters,  one  of  which  he  calls  the 
merchant's  and  the   other   the   auditor's  m  •    •    •    • 

account.     In  the  margin,  £198   19s.  lid.  • 

is  expressed  by  the  first  method,  the  low-  •  •    •    • 

est  line  being  pence,  the  second  shillings,  • 

the  third  pounds,  and  the  fourth  scores  of  •  •    •    ©    • 

pounds;  the  spaces  represent  half  a  unit  • 

of  the  next  superior  line,  and  the  detached  •    •    •    •    • 

counters  at  the  left  are  equivalent  to  five  counters  at  the  right. 
The  operations  of  addition,  subtraction,  etc.  would  be  per- 
formed in  a  manner  similar  to  those  already  given. 

The    same    sum   would    be   represented   by   the    auditor's 
account  as  in  the  margin;  the  first  group   to  the  right  being 
pence,  the  second  shillings, 
the  next  pounds,  and  the 


left     hand     group    scores 

of  pounds;  the  two  lower  x 

lines  denote  units  of  their  respective  classes,  while  in  the  third 

line  those  on  the  left  denote  one  quarter  and  on  the   right  one 

half  of  the  next  superior  class. 

The  Chinese  Computing  Table  or  Swan-Pan,  previously 
mentioned,  is  represented  by  the  accompanying  engraving. 
It  consists  of  a  small  oblong  board  surrounded  by  a  frame  or 
ledge,  and  parted  downwards  near  the  left  side  by  a  similar 
ledge.      It  is  then  divided  horizontally  by  ten  smooth    and 


PALPABLE    ARITHMETIC. 


159 


slender  rods  of  bamboo,  on  which  are  strung  two  small  balls 
of  ivory  or  bone  in  the  narrow  compart- 
ment, and  five  such  balls  in  the  wider 
compartment;  each  of  the  latter  on  the 
several  bars  denoting  one,  and  each  of  the 
former  expressing  five.  The  progressive 
bars,  descending  after  the  Chinese  manner 
of  writing,  have  their  values  increased  ten 
fold  at  each  step.  The  arrangement  here 
figured  denotes,  reckoning  downwards, 
the  number  5,804,712,063.  The  Swan- 
Pan  advances  to  the  length  of  ten  billions, 
or  a  thousand  times  further  than  the 
Roman  Abacus.  But  the  most  admirable  feature  of  the  in- 
strument is,  that  by  beginning  the  units  at  any  particular  bar 
the  decimal  subdivisions  of  the  unit  may  be  represented. 
The  Japanese  make  use  of  a  similar  instrument,  and  the 
facility  with  which  they  perform  arithmetical  operations  is 
truly  surprising. 

Several  persons  of  eminence,  during  our  own  times,  have 
advocated  the  revival  of  the  practice  of  calculation  by  means 
of  counters.  Prof.  Leslie  considers  this  method  as  better  cal- 
culated than  any  other  to  give  a  student  a  philosophical  knowl- 
edge of  the  classification  of  numbers,  and  the  theory  of  their 
notation ;  and  he  has  given,  in  great  detail,  examples  of  the 
representation  of  numbers  in  different  scales  of  notation  by 
counters,  and  of  operations  by  means  of  them. 

There  are  other  species  of  Palpable  Arithmetic,  some  of 
which  have  been  adapted  especially  for  the  use  of  blind 
people:  the  celebrated  Saunderson  invented  an  instrument 
for  this  purpose  with  which  he  is  said  to  have  worked  arith- 
metical questions  with  extraordinary  rapidity.  Arithmetical 
instruments  of  this  kind  possess  considerable  interest  and  im- 
portance from  their  use  in  lessening  the  privations  consequent 
upon  one  of  the  greatest  human  calamities. 


160 


THE    PHILOSOPHY    OF    ARITHMETIC. 


Among  other  arithmetical  machines  for  shortening  the  work 
of  calculation  or  relieving  the  operator  from  any  troublesome 
or  difficult  exercise  of  the  memory,  are  Napier's  virgulae,  or 
rods,  which  were  formerly  much  celebrated  and  generally  used. 
The  work  in  which  they  were  first  described  was  published  in 
1617,  under  the  title  of  Rabdologia.  In  the  dedication  to 
Chancellor  Seton,  he  says,  that  the  great  object  of  his  life  had 
been  to  shorten  and  simplify  the  business  of  calculation ;  and 
the  invention  of  logarithms,  which  he  had  just  promulgated, 
was  a  noble  proof  that  he  had  not  lived  in  vain.  These  virgu- 
lae, rods,  or  bones,  as  they  were  often  called,  were  thin  pieces 
of  brass,  ivory,  bone,  or  any  other  substance,  about  two  inches 
in  length  and  a  quarter  of  an  inch  in  breadth,  distributed  into 
ten  sets,  generally  of  five  each ;  at  the  head  of  each  of  these, 
in  succession,  was  inscribed  one  of  the  nine  digits  or  zero,  and 
underneath  them  in  each  piece  the  products  of  the  digit  at  the 
top  with  each  of  the  nine  digits  in  succession,  in  a  series  of 
eight  squares  divided  by  diagonals,  in  the  upper  part  of  which 
were  put  the  digits  in  the  place  of  tens,  and  in  the  lower  the 
digits  in  the  place  of  units.  In  order  to  multiply  any  two  num- 
bers together,  such  as  3469  and  574,  those  rods  are  to  be 
placed  in  contact  which  are  headed  by  the  digits  1,  3,  4,  6,  9, 
and  the  successive  products  of  the  terms  of  the  multiplier  into 
the  multiplicand  are  found  by  adding  successively  the  digit  on 
the  upper  half  of  the  square  to  the  right  to  that  in  the  lower  half 
of  the  square  to  the  left,  in  the  line  of  squares  which  are  oppo- 
site to  the  figure  of  the  multiplier  which  is  used ;  thus,  to  mul- 
tiply 3469  by  4,  we  take  the 
line  of  squares  opposite  4, 
represented  in  the  margin, 
and  the  product  is  13876, 
being  found  by  writing  6,  the  sum  of  4  and  3,  of  6  and  2,  etc., 
carrying  when  necessary.  In  case  of  division,  those  rods  are 
arranged  in  contact  which  are  headed  by  the  figures  of  the 
divisor,  and  we  are  thus  enabled  to  obtain  the  products  formed 
by  the  divisor  and  successive  terms  of  the  quotient. 


1 

3 

4 

6 

9 

4 

1  / 

\/ 

2   / 

5/ 

/2 

/6 

/4 

/6 

PALPABLE    ARITHMETIC. 


161 


6/ 
/ 4 

16 

8 

5   / 

/l 

2 

64 

8 

In  the  case  containing  these  rods,  which  Napier  calls  muZ- 
tiplicationis  prompluarium,  there  are  usually  found  also  two 
pieces  with  broader  faces,  one  consisting  of  three  longitudinal 
divisions,  and  the  other  of  four ;  one  of  which  is  adapted  to 
the  extraction  of  the  square,  and  the  other  of  the  cube  root ; 
in  the  first,  one  column  contains  the  nine  digits,  the  second 
their  doubles,  and  the  third  their  squares;  in  the  second,  the 
first   column    contains   the   digits,    the 
second  their  squares,  and  the  third  and 
fourth  their  cubes,  two  columns  being 
necessary  for  this  purpose  when 
the  cube  consists  of  three  places; 
thus,  the  last  division  but  one  in 
each  of  these  rods  is  represented 
as  in  the  margin,  the  digits  occupying  the  right-hand  column. 
In  our  times,  when  the  multiplication  table  is  so  much  more 
perfectly  learned  than  formerly,  the  eagerness  with  which  this 
invention  was  welcomed  will  excite  some  surprise,  considering 
that  its  only  object  was  to  relieve  the  memory  of  so  light  and 
trivial  a  burden;  but  it  is  in  accordance  with  some  of  the  pro- 
cesses elsewhere  noticed,  by  which  early  authors  endeavored 
to  simplify  arithmetical  operations. 

Pascal,  in  1642,  at  the  age  of  19,  invented  the  first  arith- 
metical machine,  properly  so  called.  It  is  said  to  have  cost 
him  such  mental  efforts  as  to  have  seriously  affected  his  health, 
and  even  to  have  shortened  his  days.  This  machine  was  im- 
proved afterwards  by  other  persons,  but  never  came  into  prac- 
tical use.  In  1G73,  Leibnitz  published  a  description  of  a 
machine  which  was  much  superior  to  that  of  Pascal,  but  more 
complicated  in  construction  and  too  expensive  for  its  work, 
since  it  was  capable  of  performing  only  addition,  subtraction 
multiplication  and  division.  But  these  machines  are  entirely 
eclipsed  by  those  of  Babbage  and  Scheutz.  In  1821,  Mr. 
Babbage,  under  the  patronage  of  the  British  government,  began 
the  construction  of  a  machine,  and  in  1833  a  small  portion  of  it 
was  put  together,  and  was  found  to  perform  its  work  with  the 
11 


162  THE   PHILOSOPHY   OF    ARITHMETIC. 

utmost  precision.  In  1834  he  commenced  to  design  a  still 
more  powerful  engine,  which  has  not  yet  been  constructed. 
The  expense  of  these  machines  is  enormous,  $80,000  having 
been  spent  on  the  partial  construction  of  the  first.  They  are 
designed  for  the  calculation  of  tables  or  series  of  numbers, 
such  as  tables  of  logarithms,  sines,  etc.  The  machine  pre- 
pares a  stereotype  plate  of  the  table  as  fast  as  calculated,  so 
that  no  errors  of  the  press  can  occur  in  publishing  the  result 
of  its  labors.  Many  incidental  benefits  have  arisen  from 
this  invention,  among  which  the  most  curious  and  valua- 
ble was  the  contrivance  of  a  scheme  of  mechanical  notation  by 
which  the  connection  of  all  parts  of  a  machine,  and  the  precise 
action  of  each  part,  at  each  instant  of  time,  may  be  rendered 
visible  on  a  diagram,  thus  enabling  the  contriver  of  machinery 
to  devise  modes  of  economizing  space  and  time  by  a  proper 
arrangement  of  the  parts  of  his  own  invention. 

A  machine  invented  by  G.  and  E.  Scheutz,  of  Stockholm, 
and  finished  in  1853,  was  purchased  for  the  Dudley  Observa- 
tory, at  Albany.  The  Swedish  government  paid  $20,000  as  a 
gratuity  towards  its  construction.  The  inventors  wished  to 
attain  the  same  ends  as  Mr.  Babbage,  but  by  simpler  means. 
It  can  express  numbers  decimally  or  sexagesimally,  and  prints 
by  the  side  of  the  table  the  corresponding  series  of  numbers 
or  arguments  for  which  the  table  is  calculated.  It  has  already 
calculated  a  table  of  the  true  anomaly  of  Mars  for  each  y1^  of  a 
day.  In  size,  it  is  about  equal  to  a  boudoir  piano.  Other 
attempts  have  been  made,  but  so  far  nothing  has  been  accom- 
plished which  is  entirely  satisfactory,  though  the  utility  of 
some  such  engine  in  the  calculation  of  astronomical  and  other 
tables  is  so  great,  that  it  is  quite  probable  that  efforts  will  be 
continued  until  complete  success  is  attained. 


SECTION  III. 

ARITHMETICAL  REASONING. 


I.    Thebb  is  Reasoning  in  Arithmetic. 

k 


II.    Nature  op  Arithmetical  Reasoning. 


III.    Reasoning  in  the  Fundamental  Operations, 


IV     Arithmetical  Analysis. 


V.    The  Equation  in  Arithmetic 


VI.    Induction  in  Arithmetic. 


CHAPTER  I. 

THERE   IS  REASONING   IN  ARITHMETIC. 

ALL  reasoning  is  a  process  of  comparison ;  it  consists  in 
comparing  one  idea  or  object  of  thought  with  another. 
Comparison  requires  a  standard,  and  this  standard  is  the  old, 
the  axiomatic,  the  known.  To  these  standards  we  bring  the 
new,  the  theoretic,  the  unknown,  and  compare  them  that  we 
may  understand  them.  The  law  of  correct  reasoning,  there- 
fore, is  to  compare  the  new  with  the  old,  the  theoretic  with  the 
axiomatic,  the  unknown  with  the  known. 

This  process,  simple  as  it  seems,  is  the  real  process  of  all 
reasoning.  We  pass  from  idea  to  truth,  and  from  lower  truth 
to  higher  truth,  in  the  endless  chain  of  science,  by  the  simple 
process  of  comparison.  Thus  the  facts  and  phenomena  of  the 
material  world  are  understood,  the  laws  of  nature  interpreted, 
and  the  principles  of  science  evolved.  Thus  we  pass  from  the 
old  to  the  new,  from  the  simple  to  the  complex,  from  the  known 
to  the  unknown.  Thus  we  discover  the  truths  and  principles 
of  the  world  of  matter  and  mind,  and  construct  the  various 
sciences.  Comparison  is  the  science-builder  ;  it  is  the  architect 
which  erects  the  temples  of  truth,  vast,  symmetrical,  and  beauti- 
ful. 

In  mathematics  this  process  is,  perhaps,  more  clearly  exhib- 
ited than  in  any  other  science.  In  geometry,  the  definitions 
and  axioms  are  the  standards  of  comparison  ;  beginning  in 
these,  we  trace  our  way  from  the  simplest  primary  truth  to  the 
profoundest  theorem.     In  arithmetic  we  have  the  same  basis, 

(165  J 


166  THE    PHILOSOPHY    OF    ARITHMETIC. 

and  proceed  by  the  same  laws  of  logical  evolution.  Defini- 
tions, as  a  description  of  fundamental  ideas,  and  axioms,  as  the 
statement  of  intuitive  and  necessary  truths,  are  the  foundation 
upon  which  we  rear  the  superstructure  of  the  science  of  num- 
bers. 

These  views,  though  admitted  in  respect  of  geometry,  have 
not  always  been  fully  recognized  as  true  of  arithmetic.  The 
subject,  as  presented  in  the  old  text-books,  was  simply  a  col- 
lection of  rules  for  numerical  operations.  The  pupil  learned 
the  rules  and  followed  them,  without  any  idea  of  the  reason 
for  the  operation  dictated.  There  was  no  thought,  no  deduc- 
tion from  principle;  the  pupil  plodded  on,  like  a  beast  of  burden 
or  an  unthinking  machine.  There  was,  in  fact,  as  the  subject 
was  presented,  no  science  of  arithmetic.  We  had  a  science  of 
geometry,  pure,  exact,  and  beautiful,  as  it  came  from  the  hand 
of  the  great  masters.  Beginning  with  primary  conceptions 
and  intuitive  truths,  the  pupil  could  rise  step  by  step  from  the 
simplest  axiom  to  the  loftiest  theorem  ;  but  when  he  turned  his 
attention  to  numbers,  he  found  no  beautiful  relations,  no  inter- 
esting logical  processes,  nothing  but  a  collection  of  rules  for 
adding,  subtracting,  calculating  the  cost  of  groceries,  reckoning 
interest,  etc.  Indeed,  so  universal  was  this  darkness,  that  the 
metaphysicians  argued  that  there  could  be  no  reasoning  in  the 
science  of  numbers,  that  it  is  a  science  of  intuition ;  and  the 
poor  pupil,  not  possessing  the  requisite  intuitive  power,  was 
obliged  to  plod  along  in  doubt,  darkness,  and  disgust. 

Thus  things  continued  until  the  light  of  popular  education 
began  to  spread  over  the  land.  Men  of  thought  and  genius 
began  to  teach  the  elements  of  arithmetic  to  young  pupils ; 
and  the  necessity  of  presenting  the  processes  so  that  children 
could  see  the  reason  for  them,  began  to  work  a  change  in  the 
science  of  numbers.  Then  came  the  method  of  arithmetical 
analysis,  in  that  little  gem  of  a  book  by  Warren  Colburn.  It 
touched  the  subject  as  with  the  wand  of  an  enchantress,  and 
it    began  to    glow  with  interest  and    beauty.     What    before 


THERE   IS   REASONING   IN    ARITHMETIC.  167 

was  dull  routine,  now  became  animated  with  the  spirit  of 
logic,  and  arithmetic  was  enabled  to  take  its  place  beside  its 
sister  branch,  geometry,  in  dignity  as  a  science,  and  value  as 
an  educational  agency. 

Before  entering  into  an  explanation  of  the  character  of  arith- 
metical reasoning,  it  may  be  interesting  to  notice  the  views  of 
some  metaphysicians  who  have  touched  upon  this  subject.  It 
has  been  maintained,  as  already  indicated,  by  some  eminent 
logicians,  that  there  is  no  reasoning  in  arithmetic.  Mansel 
says,  "  There  is  no  demonstration  in  pure  arithmetic,"  and  the 
same  idea  is  held  by  quite  a  large  number  of  metaphysicians. 
This  opinion  is  drawn  from  a  very  superficial  view  of  the  sub- 
ject of  arithmetic, — a  not  uncommon  fault  of  the  metaphysician 
when  he  attempts  to  write  upon  mathematical  science.  The 
course  of  reasoning  which  led  to  this  conclusion,  is  probably 
as  follows : 

First,  addition  and  subtraction  were  considered  the  two  fun- 
damental processes  of  arithmetic  ;  all  other  processes  were 
regarded  as  the  outgrowth  of  these,  and  as  contained  in  them. 
Second,  there  is  no  reasoning  in  addition ;  that  the  sum  of  2 
and  3  is  5,  says  Whewell,  is  seen  by  intuition ;  hence  subtrac- 
tion, which  is  the  reverse  of  addition,  is  pure  intuition  also ; 
and  therefore  the  whole  science,  which  is  contained  in  these 
two  processes,  is  also  intuitive,  and  involves  no  reasoning. 
This  inference  seems  plausible,  and  by  the  metaphysicians  and 
many  others  has  been  considered  conclusive. 

That  this  conclusion  is  not  only  incorrect  but  absurd,  may 
be  seen  by  a  reference  to  the  more  difficult  processes  of  the 
science.  Surely,  no  one  can  maintain  that  there  is  no  reason- 
ing in  the  processes  of  greatest  common  divisor,  least  common 
multiple,  reduction  and  division  of  fractions,  ratio  and  pro- 
portion, etc.  If  these  are  intuitive  with  the  logicians,  it  is 
very  certain  that  they  require  a  great  deal  of  thinking  on  the 
part  of  the  learner.  These  considerations  are  sufficient  to  dis- 
prove their  conclusions,  but  do  not  answer  their  arguments ;  it 


163  THE   PIIILOSOPHY   OF   ARITHMETIC. 

becomes  necessary,  therefore,  to  examine  the  matter  a  little 
more  closely. 

Whether  the  uniting  of  two  small  numbers,  as  three  and  two, 
involves  a  process  of  reasoning,  is  a  point  upon  which  it  is 
admitted  there  may  be  some  difference  of  opinion.  The  differ- 
ence of  two  numbers,  however,  may  be  obtained  by  an  infer- 
ence from  the  results  of  addition,  and,  as  such,  involves  a 
process  of  reasoning.  The  elementary  products  of  the  multi- 
plication table  are  not  intuitive  truths ;  they  are,  as  will  be 
shown  in  the  next  article,  derived,  as  a  logical  inference,  from 
the  elementary  sums  of  addition.  The  same  is  also  true  in  the 
case  of  the  elementary  quotients  in  division.  Even  admitting, 
then,  that  there  is  no  reasoning  in  addition  or  subtraction,  it 
can  clearly  be  shown  that  the  derivation  of  the  elementary 
results  in  multiplication  and  division  does  require  a  process  of 
reasoning.  Passing  from  small  numbers,  which  may  be 
treated  independently  of  any  notation,  to  large  numbers  ex- 
pressed by  the  Arabic  system,  we  see  that  we  are  required  to 
reduce  from  one  form  to  another,  as  from  units  to  tens,  etc., 
which  can  be  done  only  by  a  comparison,  and  also  that  the 
methods  are  based  upon,  and  derived  from  such  general  princi- 
ples, as  "  the  sum  of  two  numbers  is  equal  to  the  sum  of  all 
their  parts,"  etc. 

The  great  mistake,  however,  in  their  reasoning,  is  in  assum- 
ing that  all  arithmetic  is  included  in  addition  and  subtraction. 
If  it  could  be  proved  that  addition  and  subtraction,  and  the 
processes  growing  immediately  out  of  them,  contain  no  rea- 
soning, a  large  portion  of  the  science  remains  which  does  not 
find  its  root  in  these  primary  processes.  Several  divisions  of 
arithmetic  have  their  origin  in  and  grow  out  of  comparison, 
and  not  out  of  addition  or  subtraction;  and  since  comparison 
is  reasoning,  the  divisions  of  arithmetic  growing  out  of  it,  it 
is  natural  to  suppose,  involve  reasoning  processes.  Ratio,  the 
comparison  of  numbers  ;  proportion,  the  comparison  of  ratios  ; 
the  progressions,  etc.,  certainly  present  pretty  good  examples 


THERE   IS   REASONING   IN   ARITHMETIC.  169 

of  reasoning.  These  belong  to  the  department  of  pure  arith- 
metic. A  proportion  is  essentially  numerical,  as  is  shown 
in  another  place,  and  belongs  to  arithmetic  rather  than  to 
geometry.  If,  in  geometry,  the  treatment  of  a  proportion 
involves  a  reasoning  process,  as  the  logicians  will  surely 
admit,  it  must  certainly  do  so  when  presented  in  arithmetic, 
where  it  really  belongs.  It  must,  therefore,  be  admitted  that 
there  is  reasoning  in  pure  arithmetic. 

Again,  if  there  is  no  reasoning  in  arithmetic  there  is  no 
science,  for  science  is  the  product  of  reasoning.  If  we  admit 
that  there  is  a  science  of  numbers,  there  must  be  some  reason 
ing  in  the  science.  And  again,  arithmetic  and  geometry  are 
regarded  as  the  two  great  co-ordinate  branches  of  mathematics. 
Now  it  is  admitted  that  there  is  reasoning  in  geometry,  the 
science  of  extension  ;  would  it  not  be  absurd,  therefore,  to  sup- 
pose that  there  is  no  reasoning  in  arithmetic,  the  science  of 
numbers  ? 

Mansel,  as  already  quoted,  says :  "  Pure  arithmetic  contains 
no  demonstrations."  If  by  this  he  means — and  I  presume  he 
does — that  pure  arithmetic  contains  no  reasoning,  he  is 
answered  by  the  previous  discussion.  If,  however,  ne  means 
that  arithmetic  cannot  be  developed  in  the  demonstrative  form 
of  geometry — that  is,  by  definition,  axiom,  proposition,  and 
demonstration — he  is  also  in  error.  Though  arithmetic  has 
never  been  developed  in  this  way,  it  can  be  thus  developed. 
The  science  of  number  will  admit  of  as  rigid  and  systematic  a 
treatment  as  the  science  of  extension.  Some  parts  of  the  sci- 
ence are  even  now  presented  thus;  the  principles  of  ratio, 
proportion,  etc.,  are  examples.  I  propose,  at  some  future  time, 
to  give  a  complete  development  of  the  subject,  after  the  manner 
of  geometry.  The  science,  thus  presented,  would  be  a  valua- 
ble addition  to  our  academic  or  collegiate  course,  as  a  review 
of  the  principles  of  numbers.  Assuming,  then,  that  there  is 
reasoning  in  arithmetic,  in  the  next  chapter  I  shall  consider 
the  nature  of  reasoning,  as  employed  in  the  fundamental  opera- 
tions of  arithmetic. 


CHAPTER  IL 

NATURE   OP   ARITHMETICAL  REASONING. 

IN  order  to  show  the  nature  of  the  reasoning  of  arithmetic, 
a  brief  statement  of  the  general  nature  of  reasoning  will  be 
presented.  All  forms  of  reasoning  deal  with  the  two  kinds  of 
mental  products,  ideas  and  truths.  An  idea  is  a  simple  notion 
which  may  be  expressed  in  one  or  more  words,  not  forming  a 
proposition ;  —  as,  bird,  triangle,  four,  etc.  A  truth  is  the 
comparison  of  two  or  more  ideas  which,  expressed  in  language, 
give  a  proposition ;  as,  a  bird  is  an  animal,  a  triangle  is  a 
polygon,  four  is  an  even  number.  The  comparison  of  two 
ideas  directly  with  each  other,  is  called  a  judgment ;  as,  a 
bird  is  an  animal,  or  five  is  a  prime  number.  Here  five  is  one 
idea,  and  a  prime  number  is  another  idea.  Judgments  give 
rise  to  propositions ;  a  proposition  is  a  judgment  expressed  in 
words. 

Nature  of  Reasoning.  —  If  we  compare  two  ideas,  not 
directly,  but  through  their  relation  to  a  third,  the  process  is 
called  reasoning.  Thus,  if  we  compare  A  and  B,  or  B  and  C, 
and  say  A  equals  B  or  B  equals  C,  these  propositions  are 
judgments.  But  if,  knowing  that  A  equals  B,  and  B  equals 
C,  we  infer  that  A  equals  C,  the  process  is  reasoning.  Rea- 
soning may,  therefore,  be  defined  as  the  process  of  comparing 
two  ideas  through  their  relation  to  a  third.  Judgment  is  a 
process  of  direct  or  immediate  comparison ;  reasoning  is  a  pro- 
cess of  indirect  or  mediate  comparison. 

(170) 


NATURE    OF    ARITHMETICAL   REASONING.  171 

In  thus  comparing  two  ideas  through  their  relation  to  a 
third,  it  is  seen  that  we  derive  one  judgment  from  two  other 
judgments;  hence  we  may  also  define  reasoning  as  the  pro- 
cess of  deriving  one  judgment  from  two  other  judgments ; 
or  as  the  process  of  deriving  an  unknown  truth  from  two 
known  truths.  The  two  known  truths  are  called  premises,  and 
the  derived  truth  the  conclusion;  and  the  three  propositions 
together  constitute  a  syllogism.  The  syllogism  is  the  simplest 
form  in  which  a  process  of  reasoning  can  be  stated.  Its  usual 
form  is  as  follows :  A  equals  B  ;  but  B  equals  C  ;  therefore  A 
equals  C.  Here  "A  equals  B"  and  "B  equals  C"  are  the  pre- 
mises, and  "A  equals  C"  is  the  conclusion. 

The  premises  in  reasoning  are  known  either  by  intuition,  by 
immediate  judgment,  or  by  a  previous  course  of  reasoning. 
In  the  syllogism — "All  men  are  mortal ;  Socrates  is  a  man ; 
therefore,  Socrates  is  mortal" — the  first  premise  is  derived  by 
induction,  and  the  second  by  judgment.  In  the  syllogism — 
"  The  radii  of  a  circle  are  equal ;  R  and  R/  are  radii  of  a  cir- 
cle ;  therefore  II  and  R'  are  equal " — the  first  premise  is  an  intu- 
ition, and  the  second  is  a  judgment.  In  the  syllogism — "A 
equals  B,  and  B  equals  C  ;  therefore  A  equals  C" — both  pre- 
mises are  judgments. 

It  should  also  be  remarked  that  truths  drawn  from  the  first 
steps  of  the  reasoning  process,  do  themselves  become  the 
basis  of  other  truths,  and  these  again  the  basis  of  others,  and 
so  on  until  the  science  is  complete.  This  method  of  reasoning 
is  called  Discursive  (discursus)  ;  it  passes  from  one  truth  to 
another,  like  a  moving  from  place  to  place.  We  start  with  the 
simple  truths  which  are  so  evident  that  we  cannot  help  seeing 
them  ;  and  travel  from  truth  to  truth  in  the  pathway  of  science, 
until  we  reach  the  loftiest  conceptions  and  the  profoundest 
principles. 

Reasoning,  as  we  have  stated,  is  the  comparison  of  two 
ideas  through  their  relation  to  a  third;  or  it  may  be  defined  as 
the  derivation  of  one  judgment  from  two   other  judgments 


172  THE    PHILOSOPHY    OF   ARITHMETIC. 

These  two  judgments  are  not  always  both  expressed ;  indeed, 
in  the  usual  form  of  thought,  one  is  usually  suppressed ;  but 
both  are  implied,  and  may  be  supplied  if  desired  to  show 
the  validity  of  the  conclusion.  Every  truth  derived  by  a  pro- 
cess of  reasoning,  may  be  shown  to  be  an  inference  from  two 
propositions  which  are  the  premises  or  ground  of  inference, 
and  this  is  the  test  of  the  validity  of  the  truth  derived. 

There  are  two  kinds  of  reasoning,  inductive  and  deductive. 
Inductive  reasoning  is  the  process  of  deriving  a  general  truth 
from  several  particular  ones.  It  is  based  upon  the  principle 
that  what  is  true  of  the  many  is  true  of  the  whole.  Thus,  if 
we  see  that  heat  expands  many  metals,  we  infer,  by  induction, 
that  it  will  expand  all  metals.  Deduction  is  the  process  of 
deriving  a  particular  truth  from  a  general  one.  It  is  based 
upon  the  axiom,  that  what  is  true  of  the  whole  is  true  of  all 
the  parts.  Thus,  if  we  know  that  heat  will  expand  all  metals, 
we  infer,  by  deduction,  that  it  will  expand  any  particular 
metal,  as  iron. 

Mathematics  is  developed  by  the  process  of  deductive  rea- 
soning. The  science  of  geometry  begins  with  the  presentation 
of  its  ideas,  as  stated  in  its  definitions,  and  its  self-evident 
truths,  as  stated  in  its  axioms.  From  these  it  passes  by  the 
process  of  deduction  to  other  truths ;  and  then,  by  means  of 
these  in  connection  with  the  primary  truths,  proceeds  to  still 
other  truths;  and  thus  the  science  is  unfolded.  In  arithmetic, 
no  such  formal  presentation  of  definitions  and  axioms  is  made, 
and  the  truths  are  not  presented  in  the  logical  form,  as  in 
geometry.  From  this  it  has  been  supposed  that  there  is  no 
reasoning  in  arithmetic.  This  inference,  however,  is  incorrect; 
the  science  of  numbers  will  admit  of  the  same  logical  treat- 
ment as  the  science  of  space.  There  are  fundamental  ideas 
in  arithmetic  as  in  geometry;  and  there  are  also  fundamental, 
self-evident  truths,  from  which  we  may  proceed  by  reasoning 
to  other  truths.  In  this  chapter  I  shall  endeavor  to  show  the 
nature  of  the  reasoning  in  the  Fundamental  Operations  of 
Arithmetic. 


NATURE   OF   ARITHMETICAL   REASONING.  173 

Arithmetical  Ideas. — The  fundamental  ideas  of  arithmetic, 
as  given  in  the  process  of  counting,  are  the  successive 
numbers  one,  two,  three,  etc.  These  ideas  correspond  to  the 
different  ideas  of  geometry,  and  the  definitions  of  them  will 
correspond  to  the  definitions  of  geometry.  In  geometry,  we 
have  the  three  dimensions  of  extension,  giving  us  three  distinct 
classes  of  ideas,  lines,  surfaces,  and  volumes;  in  arithmetic 
there  is  only  one  fundamental  idea  of  succession,  giviDg  us 
but  one  fundamental  class  of  notions.  The-  primary  ideas  of 
arithmetic  are  one,  two,  three,  four,  five,  etc.,  which  correspond 
to  the  idea  of  line,  angle,  triangle,  quadrilateral,  pentagon, 
etc.,  in  geometry.  These  ideas  may  be  defined  as  in  the  cor- 
responding cases  in  geometry.  Thus  two  may  be  defined  as 
one  and  one;  three  as  two  and  one,  etc.;  or,  in  the  logical  form 
— three  is  a  number  consisting  of  two  units  and  one  unit. 
There  are  other  ideas  of  the  science  growing  out  of  relations, 
such  as  factor,  common  divisor,  common  multiple,  etc. 

Arithmetical  Axioms. — The  axioms  of  arithmetic  are  the 
self-evident  truths  that  relate  to  numbers.  There  are  two 
classes  of  axioms  in  arithmetic  as  in  geometry, — those  which 
relate  to  quantity  in  general,  that  is,  to  numbers  and  space;  and 
those  which  belong  especially  to  number.  Thus,  "Things 
that  are  equal  to  the  same  thing  are  equal  to  each  other,"  and 
"If  equals  be  added  to  equals  the  sums  will  be  equal,"  etc., 
belong  to  both  arithmetic  and  geometry.  In  geometry  we 
have  some  axioms  which  do  not  apply  to  numbers,  as  "All 
right  angles  are  equal,"  "A  straight  line  is  the  shortest  dis- 
tance from  one  point  to  another,"  etc.  There  are  also  axioms 
which  are  peculiar  to  arithmetic,  and  which  have  no  place  in 
geometry.  Thus,  "A  factor  of  a  number  is  a  factor  of  a  mul- 
tiple of  that  number,"  "A  multiple  of  a  number  contains  all 
the  factors  of  that  number,"  etc.  These  two  classes  of  axioms 
are  the  foundation  of  the  reasoning  of  arithmetic,  as  they  arc 
of  the  science  of  geometry. 

Arithmetical  Reasoning. —  The  reasoning  of   arithmetic  ia 


174  THE   PHILOSOPHY   OF   ARITHMETIC. 

deductive.  The  basis  of  our  reasoning  is  the  definitions  and 
axioms;  that  is,  the  conceptions  of  arithmetic,  and  the  self- 
evident  truths  arising  from  such  conceptions.  The  definitions 
present  to  us  the  special  forms  of  quantity  upon  which  we 
reason ;  the  axioms  present  the  laws  which  guide  us  in  the 
reasoning  process.  The  definitions  give  the  subject-matter  of 
reasoning ;  the  axioms  give  the  principles  which  determine  the 
form  of  reasoning,  and  enable  us  to  go  forward  in  the  discovery 
of  new  truths.  Thus,  having  defined  an  angle,  and  a  right 
angle,  we  can  by  comparison,  prove  that  "the  sum  of  the 
angles  formed  by  one  straight  line  meeting  another,  is  equal  to 
two  right  angles."  Having  the  definition  of  a  triangle,  by 
comparison  we  can  determine  its  properties,  and  the  relation 
of  its  parts  to  each  other.  So  in  arithmetic,  having  defined 
any  two  numbers,  as  four  and  six,  we  can  determine  their 
relation  and  properties ;  or  having  defined  least  common  mul- 
tiple, we  can  obtain  the  least  common  multiple  of  two  or  more 
numbers,  guiding  our  operations  by  the  self-evident  and  neces- 
sary principles  pertaining  to  the  subject. 

Axioms  in  Reasoning. — In  this  explanation  of  reasoning, 
it  is  stated  that  reasoning  is  a  process  of  comparing  two  ideas 
through  their  relations  to  a  third,  and  that  axioms  are  the  laws 
which  guide  us  in  comparing.  This  view  of  the  nature  of 
axioms  differs  from  the  one  frequently  presented.  Some  logi- 
cians tell  us  that  axioms  are  general  truths  which  contain  par- 
ticular truths,  and  that  reasoning  is  the  process  of  evolving 
these  particular  truths  from  the  general  ones.  The  axioms  of 
a  science  are  thus  regarded  as  containing  the  entire  science; 
if  one  knows  the  axioms  of  geometry,  he  knows  the  general 
truths  in  which  are  wrapped  up  all  the  particular  truths  of  the 
science.  All  that  is  necessary  for  him  to  become  a  profound 
geometer  is  to  analyze  these  axioms  and  take  out  what  is  con- 
tained in  them. 

The  incorrectness,  or  at  least  inadequacy  of  this  view  of 
the  nature  of  axioms  and  their  use  in  reasoning,  T  cannot  now 


NATURE   OF   ARITHMETICAL   REASONING.  175 

stop  to  consider.  Its  fallacy  is  manifest  in  the  extent  of  the 
assumption.  It  may  be  very  pleasant  for  one  to  suppose  that 
when  he  has  acquired  the  self-evident  truths  of  a  science,  he 
has  potentially,  if  not  actually,  in  his  mind  the  entire  science; 
such  an  expression  may  do  as  a  figure  of  speech,  but  does  not, 
it  seems  to  me,  express  a  scientific  truth  A  general  formula 
may  be  truly  said  to  contain  many  special  truths  which  may 
be  derived  from  it ;  thus  Lagrange's  formula  of  Mechanics 
embraces  the  entire  doctrine  of  the  science ;  but  no  axiom  can 
oe,  in  the  same  sense,  said  to  contain  the  science  of  arithmetic 
or  geometry. 

But  whatever  may  be  thought  of  this  view  of  the  nature 
and  use  of  axioms,  it  cannot  be  denied  that  the  explanation  of 
reasoning  which  I  have  given  is  correct.  Reasoning  is  the 
comparison  of  two  ideas  through  their  relation  to  a  third,  the 
comparison  being  regulated  by  self-evident  truths.  This  is  the 
view  of  Sir  William  Hamilton,  and  it  has  been  adopted  by  sev- 
eral modern  writers  on  logic.  Even  if  the  other  view  is  right 
— that  the  axioms  may  be  regarded  as  general  truths,  from 
which  the  particular  ones  are  evolved  by  reasoning — their 
practical  use  in  reasoning  coincides  with  the  explanation  of  the 
nature  of  the  reasoning  powers  which  I  have  presented ;  and  this 
idea  of  the  subject  will  be  found  to  be  much  more  readily  under- 
stood and  applied.  The  simpler  view  is  that  the  axioms  are 
laws  which  guide  us  in  the  comparison,  or  they  are  the  laws 
of  inference.  Thus,  if  I  wish  to  compare  A  and  B:  seeing 
that  they  are  each  equal  to  C,  I  can  compare  them  with  each 
other,  and  determine  their  equality  by  the  law  that  things 
which  are  equal  to  the  same  thing  are  equal  to  each  other. 
So,  if  I  have  two  equal  quantities,  I  may  increase  them  equally 
without  changing  their  relation,  according  to  the  law  enun- 
ciated in  the  axiom  that  if  the  same  quantities  be  added  to 
equals,  the  results  will  be  equal.  This  view  of  the  subject  of 
axioms  and  of  their  use  in  the  process  of  reasoning,  may  be 
supported   by  various  considerations,  and  will  be  found  to 


176  THE   PHILOSOPHY    OF   ARITHMETIC. 

throw  light  upon  several  things  in  logic  upon  which  writers 
are  sometimes  not  quite  clear.  In  the  following  chapter  I  shall 
apply  this  view  of  reasoning  to  the  fundamental  operatione  of 
arithmetic. 


CHAPTER  III. 

REASONING   IN   THE   FUNDAMENTAL   OPERATIONS. 

SCIENCE,  as  already  stated,  consists  of  ideas  and  truths. 
Truths  are  derived  either  by  intuition  or  reasoning.  Intu- 
itive truths  come  either  by  the  intuitions  of  the  Sense  or 
the  Reason ;  derivative  truths  by  the  discursive  process  of 
induction  or  deduction.  The  primary  ideas  of  arithmetic  are 
the  individual  numbers,  one,  two,  three;  its  primary  truths  are 
the  elementary  sums  and  differences  of  addition  and  subtrac- 
tion. How  these  primary  truths  are  derived,  is  a  question 
upon  which  opinion  is  divided.  On  the  one  hand  it  is  claimed 
that  they  are  intuitive  ;  on  the  other,  that  they  are  derived  by  rea- 
soning. Thus,  two  and  one  are  three,  three  and  two  are  Jive,  etc., 
are  regarded  by  some  as  pure  axioms,  neither  requiring  nor 
admitting  of  a  demonstration ;  while  others  regard  them  as 
deductions  from  the  primary  process  of  counting.  Let  us  ex- 
amine the  subject  somewhat  in  detail,  and  also  consider  the 
process  of  deriving  other  truths  growing  out  of  these. 

Addition. — It  is  generally  assumed  that  the  primary  sums 
of  the  addition  tables  are  axioms.  They  are  intuitive  truths 
growing  out  of  an  analysis  of  our  conceptions  of  a  number  into 
its  parts,  or  a  synthesis  of  these  parts  to  form  the  number. 
Thus,  given  the  conception  of  nine,  by  analysis  we  see  that  it 
consists  or  is  composed  of  four  and  Jive;  or  given  four  and  five, 
by  synthesis  we  immediately  see  that  it  gives  a  combination  of 
nine  units,  or  is  equal  to  nine.  This  view  is  maintained  by  some 
eminent  logicians.  "Why  is  it,"  says  Whewell,  "  that  three 
12  (177) 


178  THE  PHILOSOPnY  OF  arithmetic. 

and  two  are  equal  to  four  and  one?  Because  if  we  look  at 
live  things  of  any  kind  we  see  that  it  is  so.  The  five  are  four 
and  one  ;  they  are  also  three  and  two.  The  truth  of  our  asser- 
tion is  involved  in  our  being  able  to  conceive  the  number  five 
at  all.  We  perceive  this  truth  by  intuition,  for  we  cannot  see, 
or  imagine  we  see,  five  things,  without  perceiving  also  that  the 
assertion  above  stated  is  true." 

The  other  view  makes  counting  the  fundamental  process, 
and  derives  the  judgments  expressed  in  the  elementary  sums 
by  inference.  Thus,  the  process  of  finding  the  sum  of  Jive 
and  four  may  be  stated  as  follows: 

The  sum  of  Jive  and/our  is  that  number  which  is  four  units  after  five; 
By  counting  we  find  that  the  number  four  units  after  five  is  nine; 
Hence,  the  sum  of  Jive  and  Jour  is  nine. 

This  is  a  valid  syllogism,  and  shows  that  the  sums  might  be 
thus  obtained,  whether  they  are  actually  so  obtained  or  not. 
It  may  be  objected,  however,  that  they  can  be  obtained  only 
in  one  way ;  and  if  intuitive,  then  it  is  not  possible  to  derive 
them  by  any  process  of  reasoning.  This  does  not  necessarily 
follow,  for  we  can  often  obtain,  by  a  process  of  reasoning,  a 
truth  which  we  could  also  derive  in  some  other  way.  If  we 
discover  a  new  metal,  it  can  be  immediately  inferred  that  heat 
will  expand  it,  since  heat  expands  all  metals,  which  is  a  pro- 
cess of  deductive  reasoning.  This  truth  may  also  be  obtained 
by  direct  experiment.  Many  examples  may  be  given  to  show 
that  a  truth  may  be  derived  by  reasoning,  which  might  also 
be  derived  in  some  other  way. 

These  fundamental  truths  may  be  used  in  obtaining  the  rela- 
tions of  different  combinations  of  numbers,  and  such  an 
operation  will  be  a  process  of  reasoning.  Thus,  it  is  not  evi- 
dent to  the  learner,  neither  is  it  intuitive  with  any  one,  that  7 
plus  2  equals  4  plus  5;  or,  what  is  less  readily  seen,  that  25 
plus  37  equals  19  plus  43.  These  are  not  axioms,  since  they 
cannot  be  seen  to  be  true  without  an  examination  of  the 
grounds  of  the  relation.     The  process  of  reasoning  to  prove 


REASONING    IN    THE    FUNDAMENTAL    OPERATIONS.        179 

the  propositions  is  as  follows:  7  plus  2  equals  9;  but  4  plus  5 
equals  9;  therefore,  7  plus  2  equals  4  plus  5  ;  or,  as  Whewell 
puts  it,  thus :  7  equals  4  and  3,  therefore  7  and  2  equals  4  and 
3  and  2 ;  and  because  3  and  2  are  5,  7  and  2  equals  4  and  5. 
In  the  former  case  the  result  depends  on  the  axiom,  "  Things 
that  are  equal  to  the  same  thing  are  equal  to  each  other ;"  in 
the  latter  case,  the  reasoning  process  is  based  upon  the  axiom, 
"When  equals  are  added  to  equals  the  results  are  equal."  It 
will  be  noticed  that  Whewell's  method  of  proof  is  very  similar 
to  the  ordinary  demonstration  of  the  theorem  that  "When  one 
straight  line  meets  another  straight  line,  the  sum  of  the  two 
angles  equals  two  right  angles." 

That  this  is  a  valid  process  of  reasoning  is  evident  from  its 
similarity  to  the  geometrical  process — A  plus  13  equals  C  ;  but 
D  plus  E  equals  C  ;  therefore,  A  plus  B  equals  D  plus  K.  It 
is  readily  seen  that  many  such  cases  will  arise  in  which  the 
operations  are  entirely  independent  of  the  notation  employed, 
from  which  it  cannot  be  doubted  that  there  is  reasoning  in 
addition  in  pure  arithmetic.  When  we  proceed  to  the  addition 
of  large  numbers,  expressed  by  the  Arabic  system,  which  may 
not  be  regarded  as  pure  arithmetic,  we  base  the  operation  upon 
the  axiom  that  the  sum  of  several  numbers  is  equal  to  the  sum 
of  all  the  parts  of  those  numbers.  That  the  derivation  of  a 
result  from  this  general  axiomatic  principle  is  a  process  of  rea- 
soning, cannot  be  doubted  by  any  one  who  is  competent  to 
understand  in  what  reasoning  consists. 

Subtraction.— Subtraction,  like  addition,  embraces  two  cases, 
the  finding  of  the  difference  between  numbers  independently  of 
the  notation  employed  to  express  them,— that  is,  the  elementary 
differences  of  the  subtraction  table, — and  the  finding  of  the  dif- 
ference between  large  numbers  expressed  in  the  Arabic  system. 
The  elementary  differences  in  subtraction  may  be  obtained 
in  two  ways.  First,  we  may  find  the  difference  between  two 
numbers  by  counting  off  from  the  larger  number  as  many  units 
as  are  contained  in  the  smaller  number.     Thus,  if  we  wish   to 


180  THE   PHILOSOPHY   OF    ARITHMETIC. 

subtract  four  from  nine,  we  may  begin  at  nine  and  count  back- 
ward four  units,  and  find  we  reach  five,  and  thus  see  that 
four  from  nine  leaves  five.  The  other  method  consists  in 
deriving  the  elementary  differences  by  inference  from  the  ele- 
mentary sums.  The  former  method  is  regarded  by  some  as 
intuitive,  although  it  admits  of  a  syllogistic  statement;  the 
latter  method,  without  doubt,  involves  a  process  of  reasoning. 

To  illustrate,  suppose  we  wish  to  find  the  difference  between 
nine  and  five.  The  ordinary  process  of  thought  is  as  follows: 
Since  four  added  to  five  equals  nine,  nine  diminished  by  five 
equals  four.  This  process,  put  in  the  formal  manner  of  the 
syllogism,  is  as  follows: 

The  difference  between  two  numbers  is  a  number  which  added  to 
the  less  will  equal  the  greater  ; 

But  four  added  to  Jive,  the  less,  equals  nine,  the  greater  ; 

Therefore,  four  is  the  difference  between  nine  and  five. 

This,  of  course,  is  too  formal  for  ordinary  language,  but  is 
all  implied  in  the  practical  form,  "five  from  nine  leaves  fourr 
since  five  and  four  are  nine."  In  subtracting  large  numbers 
expressed  by  the  Arabic  system  of  notation,  we  proceed  upon 
the  principle  that  the  difference  between  the  parts  of  numbers 
equals  the  difference  between  the  numbers  themselves,  which 
shows  that  the  process  is  one  of  deduction. 

Multiplication. — Multiplication,  like  addition  and  subtrac- 
tion, embraces  two  cases — the  finding  of  the  elementary  pro- 
ducts of  the  multiplication  table,  and  the  use  of  these  in 
ascertaining  the  product  of  two  numbers  expressed  by  the 
Arabic  system.  The  elementary  products  are  obtained  by 
deduction  from  the  elementary  sums  of  addition.  Thus,  in 
obtaining  the  product  of  three  times  four,  the  logical  form  of 
thought  is  as  follows: 

Three  times  four  are  the  sum  of  three  fours  ; 
But  the  sum  of  three  fours  is  twelve; 
Uence,  three  times  four  are  twelve. 
The  first  premise  is  an  immediate  inference  from  the  defini- 


REASONING    IN    THE   FUNDAMENTAL    OPERATIONS.       181 

tion  of  multiplication ;  the  second  premise  we  know  to  be  true 
from  addition;  the  conclusion  is  a  deductive  inference  from  the 
two  premises.  In  the  common  form  of  thought  we  omit  one 
of  the  premises,  saying,  "three  times  four  are  twelve,  since 
the  sum  of  three  fours  is  twelve."  The  multiplication  of  large 
numbers  depends  on  these  elementary  products  thus  derived 
by  deduction,  and  also  employs  the  principle,  that  the  sum  of 
the  products  of  the  parts  equals  the  whole  product. 

Division. — The  reasoning  in  division  is  similar  to  that  in 
multiplication.  The  elementary  quotients  of  the  division  table 
may  be  obtained  in  two  distinct  ways — by  subtraction  or 
reverse  multiplication,  but  in  either  case  they  are  an  inference 
from  things  already  known,  and  are  thus  derived  by  a  process 
of  reasoning.  By  the  method  of  subtraction  we  say,  "  four  is 
contained  in  twelve  three  times,  since  four  can  be  subtracted 
from  twelve  three  times;  by  the  method  of  reverse  multiplica- 
tion we  say,  u four  is  contained  in  twelve  three  times,  since 
three  times  four  are  twelve.11  Each  of  these  may  be  expressed 
in  the  form  of  a  syllogism,  as  in  multiplication.  The  division 
of  larger  numbers  is  based  on  these  elementary  quotients,  and 
also  upon  the  principle  that  the  sum  of  the  partial  quotients 
equals  the  entire  quotient. 

The  view  here  given  concerning  the  origin  of  the  elementary 
products  and  quotients  may  be  presented  in  another  way. 
When  we  begin  addition  we  have  no  idea  of  multiplication  ; 
by  and  by  the  idea  of  a  product  arises  in  the  mind,  and  it  is 
immediately  seen  that  the  product  of  the  number  is  the  sum 
arising  from  talcing  one  number  as  many  times  as  there  are 
units  in  another.  Suppose  then  we  wish  to  know  the  product 
of  3  times  4,  we  reason  as  follows: 

The  product  of  3  times  4  equals  the  sum  of  4  taken  3  times; 
But  the  sum  of  4  taken  3  times  we  find  is  12 ; 
Hence,  the  product  of  3  times  4  equals  12. 

Primary  quotients  may  be  obtained  in  a  similar  manner,  and 
both  art  valid  forms  of  reasoning.     But  whatever  view  may 


182  THE    PHILOSOPHY    OF    ARITHMETIC. 

be  taken  of  the  origin  of  the  elementary  truths  of  the  funda- 
mental operations — and  the  fact  of  a  difference  of  opinion  indi- 
cates a  reason  for  it — it  certainly  cannot  be  denied,  by  one  who 
will  examine,  that  there  is  reasoning  in  the  processes  growing 
out  of  these  fundamental  operations,  and  also  in  those  which 
have  their  origin  in  comparison.  These  fundamental  judgments 
of  the  tables  of  the  four  "ground  rules"  are  committed  to 
memory,  and  are  employed  in  the  reasoning  processes  by  which 
we  derive  other  truths  in  the  science. 

Other  Forms. — As  we  leave  the  fundamental  operations, 
however,  the  processes  of  reasoning  grow  more  and  more  dis- 
tinct. As  each  new  idea  is  presented,  new  truths  arise  intui- 
tively, which  become  the  basis  for  the  derivation  of  other 
truths,  the  same  as  in  geometry.  To  illustrate,  take  the  sub- 
ject of  Greatest  Common  Divisor.  As  soon  as  the  idea  of  a 
common  divisor  is  clearly  apprehended,  several  truths  are  per- 
ceived as  growing  immediately  out  of  this  conception.  These 
truths  are  intuitively  apprehended,  and  are  the  axioms  pertain- 
ing to  the  subject.  From  these  self-evident  truths,  we  proceed 
to  other  truths  by  a  process  of  reasoning  usually  called  demon- 
stration. Thus,  in  the  subject  of  greatest  common  divisor  we 
have  these  axioms: 

1.  A  divisor  of  a  number  is  a  divisor  of  any  number  of  limes  that  num- 
ber. 

2.  A  common  divisor  of  several  numbers  is  the  product  of  some  of  the 
common  factors  of  these  numbers. 

3.  The  greatest  common  dirisor  of  several  numbers  is  the  product  of  all 
the  common  prime  factors  of  these  numbers. 

4.  The  greatest  common  divisor  of  several  numbers  contains  no  factors 
but  those  which  are  common  to  all  the  numbers. 

These  truths  are  self-evident  and  necessary,  and  are  seen 
to  be  so  as  soon  as  a  clear  idea  of  the  subject  is  attained. 
They  may  be  illustrated,  but  cannot  be  demonstrated.  They 
bear  precisely  the  same  relation  to  the  arithmetical  concep- 
tion of  greatest  common  divisor  that  the  axioms  of  geometry 


REASONING   IN    THE   FUNDAMENTAL    OPERATIONS.        183 

do  to  some  of  tbe  geometrical  conceptions.  Thus,  in  geometry, 
as  soon  as  we  have  the  conception  of  a  circle,  it  is  intuitively 
seen  that  all  the  radii  are  equal  to  each  other ;  or  that  the 
radius  is  equal  to  one-half  of  the  diameter,  etc.  Such  truths 
are  made  the  basis  of  the  reasoning  by  which  we  derive  the 
other  truths  relating  to  the  circle.  If  the  process  of  obtaining 
these  derivative  truths  in  geometry  is  regarded  as  reasoning, 
surely  the  similar  processes  in  arithmetic  are  also  reasoning. 

Having  a  clear  conception  of  the  idea  of  greatest  common 
divisor,  and  of  the  self-evident  truths  or  axioms,  belonging  to 
it,  we  are  prepared  to  derive  other  truths  relating  to  the  sub- 
ject, by  the  process  of  reasoning.  As  an  example  of  a  truth 
derived  by  demonstration,  take  the  following:  The  greatest 
common  divisor  of  two  quantities  is  a  divisor  of  their  sum 
and  their  difference. 

In  order  to  demonstrate  this  theorem,  take  any  two  numbers, 
as  20  and  12.  "We  see  that  the  greatest  common  divisor  is  4. 
We  also  know  that  20  is  5  times  4  and  12  is  3  times  4.  We 
then  reason  as  follows : 

The  sum  of  the  two  numbers  equals  5  times  4  plus  3  times  4  or  8 
times  4 ; 

But  4,  the  G.  C.  D.,  is  evidently  a  divisor  of  8  times  4 ; 

Hence,  4,  the  G.  C.  D.,  is  a  divisor  of  the  sum  of  the  two  numbers. 

In  this  syllogism  "8  times  4"  is  the  middle  term,  the  "  sum 
of  the  two  numbers"  the  major  term,  and  "  4,  the  greatest 
common  divisor,"  the  minor  term ;  and  the  syllogism  is  entirely 
valid.  In  a  similar  manner  we  may  prove  that  the  greatest 
common  divisor  is  a  divisor  of  the  difference  of  the  two  num- 
bers. The  method  of  reasoning  with  20  and  12  is  seen  to  be 
applicable  to  any  two  numbers  having  a  common  divisor; 
hence  the  truth  is  general 

It  should  be  remarked  that  a  large  portion  of  the  reasoning 
in  arithmetic  consists  in  changing  the  form  of  a  quantity,  so 
that  we  may  see  a  property  which  was  concealed  in  a  previous 
form,  and  then  inferring  that  it  belongs  also  to  the  quantity  in 


184  THE    PHILOSOPHY   OF    ARITHMETIC. 

its  first  form,  since  the  value  of  the  quantity  is  not  changed  by 
changing  its  form. 

It  is  thus  seen  that  the  science  of  arithmetic,  like  geometry, 
consists  of  ideas  and  truths;  that  some  of  these  truths  are 
self-evident,  and  others  are  derived  by  a  process  of  reasoning ; 
and  that  the  process  of  reasoning  in  the  two  sciences  is  simi- 
lar. We  proceed  now  to  consider  some  of  these  forms  of  rea- 
soning, and  especially  the  subject  of  arithmetical  analysis, 
which  will  be  treated  in  the  next  chapter. 


CHAPTER  IY. 

ARITHMETICAL    ANALYSIS. 

ARITHMETICAL  Analysis  is  the  process  of  developing 
the  relation  and  properties  of  numbers  by  a  comparison 
of  them  through  their  relation  to  the  unit.  All  numbers  con 
sist  of  an  aggregation  of  units,  or  are  so  many  times  the  single 
thing ;  and  hence  bear  a  definite  relation  to  the  unit.  This 
relation  the  mind  immediately  apprehends  in  the  conception  of 
a  number  itself.  From  this  evident  relation  to  the  unit,  all 
numbers  may  be  readily  compared  with  each  other,  and  their 
properties  and  relations  discovered.  Let  us  examine  the  pro- 
cess a  little  more  in  detail. 

Unit  the  Basis. — The  basis  of  this  analysis  is  the  Unit.  The 
Unit  is  the  primary  and  fundamental  idea  of  arithmetic.  It  is 
the  basis  of  all  numbers,  a  number  being  a  repetition  of  the 
Unit,  or  a  collection  of  units  of  the  same  kind.  The  relation  of  a 
number  to  the  Unit,  or  of  the  Unit  to  a  number,  is  consequently 
immediately  seen  from  the  conception  of  a  number  itself.  The 
collection  is  intuitively  conceived  to  be  so  many  times  the 
Unit,  or  the  Unit  such  a  part  of  the  collection.  The  import- 
ance of  the  Unit,  as  the  base  of  the  comparison  of  numbers, 
is  thus  apparent.  Integers  may  be  readily  compared  with  each 
other,  through  their  relation  to  the  fundamental  elements  out 
of  which  they  are  formed. 

A  Unit  is  one  of  the  several  things  considered ;  and,  since  a 
fraction  is  a  number  of  equal  parts  of  a  Unit,  it  is  seen  tba\ 
we  have  a  second  class  of  units  which  we  may  call  fractional 

(185) 


186  THE    PHILOSOPHY    OF    ARITHMETIC. 

units.  These  two  classes  of  units  may  be  distinguished  as  the 
Unit  and  the  fractional  unit.  A  number  of  fractional  units 
gives  rise  to  a  class  of  numbers  called  fractions.  The  same 
principle  of  comparison  obtains  in  the  comparison  of  these  as 
in  the  comparison  of  integral  numbers.  A  fractional  unit 
being  one  of  several  equal  parts  of  the  Unit,  its  relation  to  the 
latter  is  simple  and  immediately  apprehended.  We  can  thus 
compare  different  fractional  units  by  their  relation  to  the  Unit, 
as  we  did  integral  numbers  by  their  relation  to  it.  The  com- 
parison of  fractions,  which  at  first  might  have  seemed  difficult, 
thus  becomes  simple  and  easy. 

From  this  consideration  we  are  enabled  to  see  the  import- 
ance of  the  Unit  in  the  process  of  arithmetical  analysis.  As 
the  basis  of  numbers,  it  becomes  the  basis  of  reasoning  with 
numbers.  We  compare  number  with  number  or  fraction  with 
fraction  by  their  intermediate  relation  to  the  Unit.  The  Unit 
thus  becomes  the  stepping-stone  of  the  reasoning  process,  the 
central  point  around  which  the  circle  of  logic  revolves. 

Comparison  of  Integers. — Numbers  are  compared,  as  has 
already  been  remarked,  by  their  relation  to  the  Unit.  In  the 
comparison  of  numbers,  the  relation  between  them  is  not  imme- 
diately apprehended;  but  knowing  the  relation  that  each  sustains 
to  the  Unit,  we  can  ascertain  their  relation  to  each  other  by 
this  simple  intermediate  relation.  To  illustrate  this,  suppose  we 
wish  to  compare  any  two  numbers,  as  3  and  5  ;  let  the  problem 
be  "  What  is  the  relation  of  3  to  5  ?"  or  "  3  is  what  part  of  5  ?" 
We  would  reason  thus  :  One  is  1  fifth  of  5,  and  if  one  is  1  fifth 
of  5,  3,  which  is  three  times  one,  is  three  times  1  fifth,  or  3 
fifths  of  5.  Hence,  3  is  3  fifths  of  5.  In  this  example  we  cannot 
compare  3  directly  with  5;  we  therefore  make  the  comparison 
indirectly,  by  considering  their  intermediate  relation  to  the 
Unit,  which  is  readily  apprehended.  Again,  take  the  problem, 
"If  3  times  a  number  is  12,  what  is  5  times  the  number?" 
Here,  it  may  be  remarked,  3  times  the  number  is  the  known 
quantity,  and  5  times  the  number  is  the  unknown  quantity, 


ARITHMETICAL    ANALYSIS.  187 

which  we  wish  to  find  by  comparing  it  with  the  known  quan- 
tity. How  shall  we  make  this  comparison,  and  thus  pass 
from  the  known  to  the  unknown?  We  cannot  compare  them 
directly,  since  the  relation  between  them  is  not  readily  per- 
ceived ;  we  must  compare  them  indirectly  by  means  of  their 
relation  to  the  Unit.  The  process  of  reasoning  is  as  follows: 
If  3  times  a  number  is  12,  once  the  number  is  ^  of  12  or  4; 
and  if  once  a  number  is  4,  five  times  the  number  is  5  times  4, 
or  20.  Thus  we  readily  pass  from  three  times  the  number  to 
five  times  the  number — from  the  known  to  the  unknown — first 
passing  from  three  to  one  and  then  from  one  to  five.  In  the 
same  manner  all  numbers  may  be  compared  with  each  other, 
their  relation  being  determined  by  this  intermediate  relation  to 
One,  the  Unit,  the  basis  of  all  numbers. 

Comparison  of  Fractions. — Fractions  are  also  compared  by 
means  of  their  relation  to  the  Unit.  A  Fraction  is  a  number 
of  fractional  units.  The  fractional  unit  is  one  of  several 
equal  parts  of  the  Unit;  hence  the  relation  between  it  and  the 
Unit  is  simple  and  readily  perceived.  When  we  have  a  num- 
ber of  fractional  units — that  is,  a  Fraction — in  comparing  it 
with  the  Unit,  we  must  first  pass  from  the  number  of  fractional 
units  to  the  fractional  unit  itself,  and  then  from  the  fractional 
unit  to  the  Unit.  From  this  we  can  readily  pass  to  a  num- 
ber, or  to  any  other  fractional  unit,  and  then  to  any  number 
of  such  fractional  units,  that  is,  to  any  fraction.  This  will  be 
more  clearly  seen  by  its  application  to  a  problem. 

Take  the  problem,  "  If  -§  of  a  number  is  24,  what  is  f  of  the 
number?"  We  reason  thus:  If  to>-thirds  of  a  number  is  24, 
one-third  of  the  number  is  \  of  24,  or  12 ;  and  Mree-thirds,  or 
once  the  number,  is  3  times  12,  or  36.  If  once  the  number  is 
36,  one-fourth  of  the  number  is  \  of  36,  or  9;  and  £nree-fourths 
of  the  number  is  3  times  9,  or  27.  In  this  problem  we  compare 
the  two  fractions  §  and  f ,  by  passing  from  fr/w-thirds  down  to 
one-third,  then  rising  up  to  the  Unit,  then  passing  down  to  one- 
fourth,  and  then  up  to  ^/iree-fourths.     In  other  words,  we  pass 


188 


THE    PHILOSOPHY    OF    ARITHMETIC. 


relation  of  f  to  f  ? 

which  it  is  required  to  compare  -g 


from  a  number  of  fractional  units  to  the  fractional  unit,  then 
to  the  Unit,  then  to  another  fractional  unit,  and  then  to  a 
number  of  those  fractional  units.  We  first  go  down,  then  up, 
uhen  down  again,  and  then  up  again  to  the  required  point. 

Another  excellent  example  of  this  method  of  comparison  is 
given  in  the  solution  of  the  following  problem:  What  is  the 
Here  -J  is  the  basis  of  comparison  with 
f.  This  relation  cannot  be 
immediately  seen,  but  it  can  readily  be  determined  by  the 
method  of  analysis.  The  solution  is  as  follows:  One-fifth  is  \ 
of  f,  and  if  one-fifth  is  \  of  f,  ^t-e-fifths,  or  One,  is  5  times  \ 
or  j  of  f.  If  One  is  f  of  f ,  one-third  is  \  of  f  or  ^  of  f ,  and 
ftt'o-thirds  is  2  times  -fa,  or  f ;  hence  f  is  £  of  -f.  In  this  prob- 
lem we  see  the  same  law  of  comparison,  and  this  law  runs 
through  the  entire  subject. 

Having  given  this  general  idea  of  the  process,  I  will  state 
the  several  simple  cases  of  arithmetical  analysis,  and  illustrate 
the  process  of  thought  by  means  of  a  diagram.  The  central 
relation  of  the  Unit  to  the  thought  process,  and  the  transition 
from  the  Unit  and  to  the  Unit,  will  be  readily  seen. 

Case  I. —  To   pass  from    the   Unit    to  B 

any  number.  Take  the  problem :  If  1 
apple  costs  3  cents,  what  will  4  apples 
cost?  If  1  apple  costs  3  cents,  4  apples, 
which  are  4  times  1  apple,  will  cost  4  times 
3  cents,  or  12  cents.  In  this  problem  the 
mind  starts  at  the  Unit  A,  and  ascends  4  iJ»n- 
steps  to  B. 

Case  II. —  To  pass  from  any  number  to  B 
the  Unit.  Take  the  problem:  If  4  apples 
cost  12  cents,  what  will  1  apple  cost?  The 
solution  is  as  follows:  If  4  apples  cost  12 
cents,  1  apple,  which  is  1  fourth  of  4  apples, 
will  cost  1  fourth  of  12  cents,  or  3  cents. 
in  this  problem  the  mind  starts  at  the  num- 


Unlc. 


ARITHMETICAL   ANALYSIS.  189 

ber  4,  four  steps  above  the  basis,  and  steps  down  to  the  Unit, 
or  basis  of  numbers. 

Case  III. —  To  pass  from  a  number  to  a  number.  Take  the 
problem:  If  3  apples  cost  15  cents,  what  will  4  apples  cost? 
The  solution  is :  If  3  apples  cost  15  cents,  1  apple  will  cost 
i  of  15  cents,  or  5  cents,  and  4  apples  will  cost  4  times  5  cents, 
or  20  cents.  In  this  case  we  are  to  pass  from  the  collection 
three  to  the  collection  four.  In  comparing  three  and  four, 
their  relation  is  not  readily  seen  ;  but  knowing  the  relation  of 
three  to  the  Unit,  and  of  the  Unit  to  four,  we  make  the  transi- 
tion from  three  to  four  by  passing  through  the  Unit.  This 
may  be  illustrated  as  follows:  Suppose  one  standing  at  A  and 
wishing  to  pass  over  to  C. 
Unable  to  step  directly  from 
A  to  C,  he  first  steps  down  to 
the  starting  point,  B,  and  then 
ascends  to  C.  So  in  compar 
ing  numbers,  when  we  cannot 
pass  directly  from  the  one  to  '  unit. 

the  other,  we  go  down  to  the 

Unit,  or  starting-point  of  numbers,  and  then  go  up  to  the  other 
number.  These  relations  are  intuitively  apprehended,  being 
presented  in  the  formation  of  numbers.  In  the  given  problem 
we  stand  three  steps  above  the  Unit,  and  we  wish  to  go  four 
steps  above  the  Unit.  To  do  this  we  first  descend  the  three 
steps,  and  then  ascend  the  four  steps. 

Case  IV. —  To  pass  from  a  unit  to  a  fraction.  Take  the 
problem :  If  one  ton  of  hay  cost  $8,  what  will  f  of  a  ton  cost  ? 
The  solution  is  as  follows:  If  one  ton  of  hay  costs  $8,  one-fourth 
of  a  ton  will  cost  \  of  $8,  or  $2,  and  three-fourths  of  a  ton 
will  cost  3  times  $2,  or  $6. 

In  this  problem  we  pass  from  the  Unit  to  the  fourth,  one  of 
the  equal  divisions  of  the  unit,  and  then  to  a  collection  of  such 
equal  divisions.     In  other  words,  we  descend  from  the  integral 


A 

n 

B 

190 


THE    PHILOSOPHY    OF    ARITHMETIC. 


Unit. 

C 

1 

I 

B 

Unit   to   the  fractional    Unit,    and    then   ascend   among  the 

fractional   units.      It   is   as 

if  we  were  standing  at   A, 

and  wished  to  pass  to    C  ; 

we    first    take   four    steps 

down  to  B,  and  then  three 

steps  up   to    C,    instead   of 

trying  to   step  immediately  4th. 

over  from  A  to  C. 

Case  V. —  To  pass  from  a  fraction  to  a  unit.  Take  the 
problem:  Iff  of  a  ton  of  hay  cost  $6,  what  will  one  ton 
cost?  The  solution  is  as  follows:  If  //iree-fourths  of  a  ton  of 
hay  cost  $6,  one-fourth  of  a  ton  will  cost  ^  of  $6,  or  $2 ;  and 
/owr-fourths  of  a  ton,  or  one  ton,  will  cost  4  times  $2,  or  $8. 

In  this  problem  we  pass 
from  a   collection   of  frac-  c 

tional  units  to  the  frac- 
tional unit,  and  then  to 
the  integral  Unit.  It  is  as 
if  we  were  standing  at  A, 
and  wished  to  pass  to  C. 
We  cannot  make  the  transi- 
tion directly,  so  we  step  three  steps  down  to  B,  and  then  four 
steps  up  to  C. 

Case  VI. —  To  pass  from  a  fraction  to  a  fraction. — Take 
the  problem  •  if  |  of  a  number  is  15,  what  is  $  of  the 
number?  The  solution  is  as  follows:  If  £7i  ree-fourths  of  a 
number  is  15,  one-fourth  of  the  number  is  ^of  15  or  5,  and 
/owr-fourths  of  the  number,  or  once  the  number,  is  4  times  5 
or  20.  If  the  number  is  20,  one-fifth  of  the  number  is  1  fifth 
of  20,  or  4  ;  and  /owr-fifths  of  the  number  is  4  times  4  or  1G. 

In  this  problem  we  wish  to  compare  the  two  fractions  f  and 
|;  but  since  we  cannot  perceive  the  relation  of  them  directly, 
we  must  compare  them  through  their  relation  to  the  Unit.  To 
do  this  we  first,  go  from  ^Aree-fourths  to  one-fourth,  then  from 


A                                                (Unit. 

_r 

!  b  ! 

arithmetical  analysis.  191 

one-fourth  to  the  Unit,  then  from  the  Unit  to  one-fifth,  and 
then  to  four-fifths.  In  other  words,  we  first  go  down  from 
the  collection  of  fractional  units  to  the  fractional  Unit  and  then 
up  to  the  integral  Unit ;  we  then  descend  to  the  other  frac- 
tional unit,  and  then  ascend  to  the  number  of  fractional  units 
required.  It  is  as  if  we  were  standing  at  A  and  wished  to 
pass  to  E  ;   we  cannot  step  directly  over  from  one  point  to  the 

c 

Unit. 


d 


other  so  we  pass  from  A  three  steps  down  to  B,  then  four 
steps  up  to  C,  then  five  steps  down  to  D,  and  then  four  steps 
up  to  E. 

These  diagrams,  it  is  believed,  present  a  clear  illustration  of 
the  subject,  and  enable  one  to  understand  the  process  of  thought 
in  the  elementary  operations  of  arithmetical  analysis.  The 
Unit  is  thus  seen  to  lie  at  the  basis  of  the  process,  the  mind 
running  to  it  and  from  it  in  the  comparison  of  numbers.  It 
will  be  remembered,  however,  that  these  are  merely  illustra- 
tions, and  are  not  designed  to  convey  a  complete  idea  of  the 
process  in  all  of  its  details.  This  can  only  be  seen  by  a  care- 
ful analysis  of  the  process  itself. 

Analysis  Syllogistic. — The  process  of  arithmetical  analysis 
is  a  process  of  mediate  comparison,  and  is  consequently  a 
reasoning  process.  This  will  appear  from  the  fact  that  it  may 
be  presented  in  the  syllogistic  form.  Take  the  simplest  case: 
If  4  apples  cost  12  cents,  what  will  5  apples  cost?  Expressed 
in  the  form  of  a  syllogism,  we  have  the  following: 

The  cost  of  1  apple  is  £  of  the  cost  of  4  apples; 

But  £  of  the  cost  of  4  apples  is  \  of  12  cents,  or  3  cents ; 

llence  the  cost  of  1  apple  is  3  cents. 


192  THE  PHILOSOPHY   OF   ARITHMETIC. 

The  cost  of  five  apples  is  5  times  the  cost  of  1  apple; 

But,  5  times  the  cost  of  1  apple  is  5  times  3  cents,  or  15  cents; 

Hence,  the  cost  of  five  apples  is  15  cents. 

It  is  thus  seen  that  the  process  of  analysis  is  purely  syllo- 
gistic, and  is,  consequently,  a  reasoning  process.  It  is  not 
csuaiiy  presented  in  the  syllogistic  form,  since  it  would  be  too 
stiff  and  formal,  and  moreover  would  be  more  difficult  for  the 
young  pupil  to  understand. 

Direct  Comparison. — The  comparison  of  numbers,  so  far  as 
explained,  is  indirect  and  mediate, that  is, through  their  relation 
to  the  Unit.  After  becoming  familiar  with  this  process,  the 
mind  begins  to  perceive  the  relations  between  numbers  them- 
selves, and  is  thus  enabled  to  reason  by  comparing  the  numbers 
directly,  instead  of  employing  their  intermediate  relations  to 
the  common  basis.  To  illustrate,  take  the  problem :  If  3 
apples  cost  10  cents,  what  will  6  apples  cost?  We  may  reason 
thus:  If  3  apples  cost  10  cents,  6  apples,  which  are  two  times 
3  apples,  will  cost  two  times  10  cents,  or  20  cents.  Primarily 
we  would  have  gone  to  the  Unit,  finding  the  cost  of  one  apple ; 
but  now  we  may  omit  this  and  compare  the  numbers  directly. 

With  integral  numbers  this  direct  comparison  is  simple  and 
easy ;  but  with  fractions  it  is  much  more  complicated  and  diffi- 
cult. Thus,  if  f  of  a  number  is  20,  it  is  difficult  to  see  directly 
that  |  of  the  number  is  -f  of  20 ;  that  is,  that  the  relation  of 
*  to  §  is  f ;  hence,  though  we  should  avail  ourselves  of  the 
direct  relation  of  integral  numbers,  it  will  be  found  much  sim- 
pler to  compare  fractions  by  their  intermediate  relations  to  the 
Unit. 


CHAPTER  V. 

THE    EQUATION   IN    ARITHMETIC. 

TIIE  comparison  of  mathematical  quantities  is  mainly  con- 
cerned with  the  relations  of  equality.  The  relation  of 
equality  gives  rise  to  the  Equation,  one  of  the  most  important 
instruments  of  mathematical  investigation.  The  Equation 
lies  at  the  basis  of  mathematical  reasoning ;  it  is  the  key  with 
which  we  unlock  its  most  hidden  principles ;  the  instrument 
with  which  we  develop  its  profoundest  truths.  The  equation 
is  a  universal  form  of  thought,  and  is  not  restricted  to  any  one 
branch  of  mathematics.  In  its  simple  form  it  belongs  to 
arithmetic  and  geometry,  as  well  as  to  algebra.  The  simplest 
process  of  arithmetic,  one  and  one  are  two  (1  +  1  =  2),  is  really 
an  equation,  as  much  as  x2+ax=b. 

In  the  higher  departments  of  the  subject  of  arithmetic,  the 
uquational  form  of  thought  and  expression  becomes  indispensable. 
Much  of  the  reasoning  of  arithmetic,  which  is  not  formally 
thus  expressed,  may  be  put  in  the  form  of  the  equation.  As 
an  example,  take  the  question,  "  If  f  of  a  number  is  24,  what 
is  the  number?"  The  solution  of  this  maybe  expressed  as 
follows:  Since  §  of  the  number  =  24,  £  of  the  number  =12, 
and  f  of  the  number=36.  Here,  "the  number"  is  the  un- 
known quantity,  which  is  ascertained  by  comparing  it  with  the 
known  quantity,  24 ;  and  then,  by  the  analysis,  passing  from 
two-thirds  of  the  number  to  once  the  number.  The  illustra- 
tion given  is  of  a  very  simple  case,  but  the  same  principle 
13  (  193 ) 


194  THE    PHILOSOPHY    OF    ARITHMETIC. 

holds  in  the  most  complicated  processes  of  arithmetical  analy- 
sis. If,  instead  of  a  number,  we  had  value,  cost,  weight, 
labor,  etc.,  the  method  of  comparison  and  analysis  would  be 
the  same.  We  can  thus  see  the  use  of  the  equation,  the  great 
instrument  of  analysis,  even  in  the  elementary  processes  of 
arithmetic.  Here  it  begins  that  wondrous  career  which  ends 
in  the  deepest  analysis  and  the  broadest  generalization.  Here 
we  find  the  germ  of  that  power  which,  in  its  higher  develop- 
ment, comprehends  the  whole  science  of  Mechanics  in  a  single 
formula,  thus  holding,  potentially,  in  its  mighty  grasp,  the 
mathematical  laws  of  the  universe. 

The  equation  in  arithmetic  assumes  several  different  forms. 
We  begin  by  comparing  quantities — the  comparison  of  equal 
quantities  giving  an  equation.  A  comparison  of  unequal  quan- 
tities gives  us  ratio,  and  a  comparison  of  equal  ratios  gives  us 
another  kind  of  equation,  an  equation  of  relations,  usually  called 
a  proportion.  The  proportion  4 : 2 : :  6 : 3,  is  in  reality  an  equa- 
tion, as  much  so  as  2=2,  for  it  really  means  4-j-2=6-r-3.  The 
treatment  of  the  equation  gives  rise  to  several  special  forms  of 
logical  procedure,  such  as  transposition,  elimination,  etc. 

The  equation,  I  have  said,  belongs  to  arithmetic ;  and  this 
thought  I  desire  to  impress.  The  equation  is  a  formal  compar- 
ison of  two  equal  quantities.  This  comparison  is  being  made 
continually;  all  of  our  reasoning  involves  it;  we  cannot  think 
without  it ;  hence,  the  equation  must  enter  into  the  reasoning 
of  arithmetic.  We  compare  one  thing  with  another,  the 
known  with  the  unknown,  and  thus  attain  to  new  truths ;  and 
all  such  forms  of  comparison  involve  the  equation,  and  are 
only  possible  by  means  of  it.  The  simplest  arithmetical  pro- 
cess>  1  _f- 1  =  2,  is  as  much  an  equation  as  Du=6u-\- du,  though 
the  latter  may  express  one  of  the  profoundest  generalizations 
to  which  the  human  mind  has  attained. 

Substitution. — A  prominent  element  of  arithmetical  reason- 
ing, accompanying  the  equation,  is  substitution.  By  this  we 
mean  the  using  of  one  quantity  in  place  of  another,  to  which 


THE    EQUATION    IN    ARITHMETIC.  195 

it  is  equal.  The  object  of  this  is  that  if  we  have  an  expression 
consisting-  of  a  combination  of  several  different  quantities,  and 
know  the  relation  of  these  quantities,  we  may  so  substitute 
their  values  that  the  expression  for  the  combination  may  be 
obtained  in  terms  of  one  single  quantity,  the  value  of  which 
may  much  more  readily  be  determined;  and  then  the  values 
of  the  other  quantities,  from  their  relation  to  this  quantity, 
may  also  be  found. 

To  illustrate,  suppose  we  have  the  two  conditions,  twice  a 
number  plus  three  times  another  number  equals  48,  and  three 
times  this  second  number  equals  four  times  the  first.  We 
can  readily  solve  this  by  substituting  for  one  of  these  numbers 
its  value  in  terms  of  the  other,  thus  obtaining  a  number  of 
times  a  single  quantity,  equal  to  the  known  quantity  48.  The 
operation  maybe  exhibited  thus: 

2  times  the  tirst  number  -f  3  times  the  second  —  48; 
but,  3  times  the  second  number  =  4  times  the  first  number; 

hence,  2  times  the  tirst  number  -f  4  times  the  first  number  —  48  ; 
or,  6  times  the  first  number  =  48, 

and,  once  the  first  number  =  8, 

and  from  this  we  may  easily  find  the  second  number. 

Substitution  is  a  form  of  deductive  reasoning,  as  may  be 
seen  by  an  analysis  of  the  process.  Take  the  simple  example, 
A-f  B  — 24,  and  B  =  3A.  We  usually  reason  as  follows:  If 
A  +  B  =  24,  and  B=3A,  then  A-f3A  =  24,  or  4A  =  24,  etc. 
That  the  logical  character  of  the  process  may  appear,  we 
should  reason  thus:  If  B  — 3A,  A-4-B  will  equal  A-f  3A,  from 
the  axiom,  "  If  equals  be  added  to  equals  the  sums  will  be 
equal."  And  since  A  +  B=24,  and  A-f  B= A-f  3 A,  A-f  3 A 
must  equal  24,  from  the  axiom,  "  Things  that  are  equal  to  the 
same  thing  are  equal  to  each  other."  Substitution  is  thus 
seen  to  be  strictly  a  deductive  process.  In  practice  these  log- 
ical steps  are  omitted  for  brevity  and  conciseness,  the  argument 
being  sufficiently  clear  to  be  readily  understood. 

Substitution   is   almost  an  essential   accompaniment  of  the 


196  THE    PHILOSOPHY    OF    ARITHMETIC. 

equation.  The  comparison  of  two  equal  quantities  without 
some  other  truth,  would  often  be  of  little  value  in  attaining 
new  truth.  By  substituting  one  value  for  another,  we  can 
often  so  change  the  equation  that  it  expresses  a  relation  which 
will  immediately  lead  to  some  new  relation  of  the  known  to 
the  unknown,  by  which  we  can  attain  to  the  value  of  the  un- 
known. Substitution  has  been  supposed  to  be  restricted  to 
algebraic  reasoning;  but  this  is  not  correct.  It  is  extensively 
employed  in  geometrical  reasoning,  and  is  just  as  appropriate 
in  arithmetic  as  in  algebra. 

Transposition. — In  the  equational  form  of  thought,  so  con- 
stantly recurring  in  arithmetic,  it  sometimes  occurs  that  we 
have  a  multiple  of  a  quantity  compared  with  another  multiple 
of  the  same  quantity,  increased  or  diminished  by  some  other 
quantity.  In  such  cases  it  is  natural  to  desire  to  unite  these 
two  multiples  into  one,  which  is  done  by  so  changing  them  as 
to  bring  them  on  the  same  side  of  the  equation.  This  is  what 
is  known  as  transposition.  It  is  consequently  seen  that  trans- 
position is  a  process  not  foreign  to  arithmetic,  but  one  entirely 
legitimate  and  natural  in  the  comparison  of  arithmetical  ideas. 

Other  processes  of  thought  analogous  to  those  which  occur 
in  algebra  are  employed  in  arithmetical  reasoning.  The  mind 
here  takes  the  first  step  in  equational  thought,  which,  when 
generalized,  leads  it  to  the  high  altitudes  of  mathematical  sci- 
ence. Ilere  it  plumes  its  wings  to  follow  the  master  minds  in 
their  lofty  flights  in  a  region  of  thought  far  beyond  that  of 
which  the  mere  arithmetician  could  even  dream.  The  object 
of  this  chapter  is  not  to  give  a  philosophical  discussion  of  the 
equation  in  general,  but  to  show  that  it  has  a  place  even  in 
arithmetical  reasoning,  which  has  sometimes  been  doubted  or 
denied. 


CHAPTER  VI. 

INDUCTION    IN    ARITHMETIC. 

MATTIEMATICS  is  a  deductive  science,  and  all  of  its 
truths,  not  axiomatic,  may  be  derived  by  a  deductive  pro- 
cess of  reasoning.  Is  it  possible,  however,  to  obtain  any  of 
these  truths  by  Induction?  This  is  a  disputed  question;  it 
will  therefore*  it  is  thought,  be  of  interest  to  enter  somewhat 
into  details  in  its  discussion.  I  believe  it  can  be  shown  that, 
there  are  many  truths  in  mathematics  that  can  be  proved  by 
induction;  and,  furthermore,  that  many  of  its  truths  were 
originally  obtained  by  an  inductive  process;  and  still  further, 
that  induction  is,  in  many  cases,  a  legitimate  method  of  math- 
ematical investigation. 

Induction,  as  is  generally  known,  is  a  process  of  thought 
from  particular  facts  and  truths  to  general  ones.  It  is  the 
logical  process  of  inferring  a  general  truth  from  particular  facts 
or  truths.  Thus,  if  I  observe  that  heat  will  expand  the  sev- 
eral metals,  iron,  tin,  zinc,  lead,  etc.,  I  may  infer,  since  these 
are  representatives  of  the  class  of  metals,  that  heat  will  ex- 
pand all  metals.  It  is  thus  seen  to  be  a  process  of  reasoning, 
based  upon  the  principle  that  what  is  true  of  the  individuals 
is  true  of  the  class.  The  basis  of  Induction  is  the  general 
proposition  that  what  is  true  of  the  many  is  true  of  the  wnole; 
or,  as  Esser  states  it,  "  What  belongs  or  does  not  belong  to 
many  things  of  the  same  kind  belongs  or  does  not  belong  to 
all  things  of  the  same  kind." 

That  this  method  of  reasoning  can  be  employed  in  arithme- 

(197) 


iyy  IHE    PHILOSOPHY    OF    ARITHMETIC. 

tic  appears  evident  a  priori.  It  is  certainly  not  unreasonable 
to  suppose  that  we  may,  upon  finding  a  truth  which  holds  in 
several  particular  cases  in  arithmetic,  infer  that  it  will  hold 
good  in  all  similar  cases.  This  conclusion  is  strengthened  by 
the  fact  that  arithmetic  is  somewhat  special  in  its  nature,  par- 
ticularly so  as  compared  with  algebra.  Its  symbols  represent 
special  numbers,  and  dealing  thus  with  special  symbols,  it  is 
to  be  expected  that  we  would  discover  some  truths  which  hold 
in  particular  instances,  before  we  know  of  their  general  applica- 
tion. That  it  is  not  only  possible  to  reason  inductively  in 
arithmetic,  but  that  we  do  reason  thus,  may  be  shown  by  act 
ual  examples. 

First,  take  the  property  of  the  divisibility  of  numbers  by 
nine.  Suppose  that,  not  knowing  this  property,  I  divide  a 
number  by  9.  and  then  divide  the  sum  of  the  digits  by  9,  and 
thus  see  that  both  remainders  are  the  same.  Suppose  I  should 
try  this  with  several  different  numbers,  and  seeing  that  it  holds 
srood  in  each  case  infer  that  ii  is  true  in  all  cases:  should  1 
not  have  entire  faith  in  my  conclusion,  and  would  not  this 
inference  be  well  founded?  This  is  an  inductive  inference, 
and  is  as  legitimate  as  the  inference  that  heat  expands  all 
metals,  because  we  see  that  it  expands  the  several  particular 
metals,  iron,  zinc,  tin,  etc. 

Second,  take  a  number  of  two  digits,  as  37  ;  invert  the 
digits, and  take  the  difference  between  the  two  numbers,  and 
we  have  73  —  37  equal  to  36,  in  which  the  sum  of  the  two 
digits,  3  and  6,  equals  9.  If  we  take  several  other  numbers  of 
two  digits  and  do  the  same,  we  shall  find  the  sum  of  the  two 
digits  to  be  also  9;  and  observing  that  this  is  true  in  several 
cases,  we  may  infer  that  it  is  true  in  all  cases,  in  which  we 
again  have  a  true  inductive  inference. 

Third,  take  a  proportion  in  arithmetic,  and,  by  actual  mul- 
tiplication, we  shall  see  that  the  product  of  the  means  equals 
the  product  of  the  extremes.  Examining  several  proportions, 
we  shall  see  that  the  same  is  true  in  each  case,  and  from  these 


INDUCTION    IN    ARITHMETIC.  199 

we  can  infer  that  it  is  true  in  all  eases,  in  which  we  again 
arrive  at  a  general  truth  by  induction.  This  is  not  only  legit- 
imate inference,  but  it  is  actually  the  way  in  which  pupils 
naturally  derive  the  truth  before  they  understand  how  to 
demonstrate  it. 

Now,  of  course  each  of  the  above  principles  will  admit  of 
rigorous  demonstration  by  deduction;  what  I  hold,  and  what 
I  think  is  clearly  shown,  is,  that  they  can  also  be  derived  by 
induction.  Deduction  would  prove  that  they  must  be  so  ;  in- 
duction merely  shows  that  they  are  so.  Many  other  examples 
from  arithmetic  might  be  given  in  illustration  of  the  same 
thing.  But  the  use  of  induction  in  mathematics  is  not  con- 
fined to  arithmetic;  if  we  go  to  algebra  we  shall  find  that  the 
same  method  of  reasoning  may  be,  and  indeed  is,  employed 
there.  The  theorem,  xn—yn  is  divisible  by  x— y,  may  be 
proved  by  pure  induction.  Try  the  several  cases  xl — y2, 
x* — y3,  x* — y4,  etc.,  and  seeing  that  the  division  is  exact  in  the 
several  cases,  it  is  entirely  legitimate  to  infer  that  it  will  be 
exact  in  all  similar  cases,  or  that  xn—yn  is  divisible  by  x—y. 
The  same  thing  may  be  shown  in  many  other  cases,  but  it  is 
needless  to  multiply  examples.  Even  in  geometry  the  same 
method  may  be  applied.  I  knew  a  young  person  who,  before 
he  studied  geometry,  derived  by  trial  and  induction  the  fact 
that  there  may  be  a  series  of  right-angled  triangles,  whose 
sides  are  in  the  proportion  of  3,  4,  and  5 ;  and  there  is  no  doubt 
that  the  ancients  knew  that  the  square  of  the  hypothenuse 
equaled  the  sum  of  the  squares  on  the  other  two  sides,  long 
before  Pythagoras  demonstrated  it. 

I  have  said  that  some  of  the  truths  of  mathematics  were 
discovered  by  induction;  among  these  the  most  prominent, 
perhaps,  is  Newton's  Binomial  Theorem.  Newton  discovered 
this  theorem  by  pure  induction.  He  left  no  demonstration  of 
it,  and  yet  it  was  considered  of  so  much  importance  that  it 
was  engraved  upon  his  tomb.  His  first  principles  of  Calculus 
were  somewhat  inductive  in  their  origin,  as  may  be  seen  in 
his  Principia. 


200  THE    PHILOSOPHY    OF    ARITHMETIC. 

The  following-   formula  is  used  for  finding  the  number  of 
primes  up  to  the  number  x,  when  x  is  a  large  number: 


N=- 


A  log  x—B  ' 

in  which  N  denotes  the  number  of  primes,  and  A  and  B  are 
constants  to  be  determined  by  trial.  This  formula  was 
derived  by  a  process  of  induction.  It  is  found  to  satisfy  the 
tables  of  prime  numbers,  but  no  deductive  demonstration  of  it 
has  yet  been  given,  and  it  must  therefore  be  regarded  as  empir- 
ical. 

In  the  theory  of  numbers  we  have  the  following  remarkable 
property:  Every  number  is  the  sum  of  one,  two,  or  three 
triangular  numbers;  the  sum  of  one,  two,  three,  or  four 
square  numbers ;  the  sum  of  one,  two,  three,  four,  or  Jive 
pentagonal  numbers,  and  so  on.  This  law,  though  known  to 
be  entirely  general,  has  never  been  demonstrated  except  for 
the  triangular  and  square  numbers.  It  was  discovered  by 
Fermat,  who  intimates,  in  his  notes  on  Diophantus,  that  he 
was  in  possession  of  a  demonstration  of  it;  which,  however,  is 
doubtful,  since  such  mathematicians  as  Lagrange,  Legendre, 
and  Gauss  have  failed  to  demonstrate  it.  The  general  law  is 
at  present  accepted  on  the  basis  of  induction. 

It  is  thus  clearly  seen  that  many  of  the  truths  of  mathemat- 
ics can  be  derived  by  induction;  that  is,  by  inferring  general 
truths  from  particular  cases.  It  is  not  claimed,  however,  that 
this  changes  the  nature  of  the  science.  I  have  before  said 
that  mathematics  is  a  deductive  science ;  my  object  has  been 
merely  to  show  the  error  of  those  who  hold  that  it  is  impos- 
sible to  derive  any  of  the  truths  of  mathematics  by  induction. 

I  have  called  especial  attention  to  this  subject,  on  account 
of  the  obscure  and  conflicting  views  which  seem  to  exist  con- 
cerning it.  Several  authors  speak  of  the  inductive  methods  of 
treating  arithmetic,  while  others  as  positively  assert  that  there 
can  be  no  inductive  treatment  of  the  science.  The  logicians 
lead   us  to  infer  that    induction  cannot  be  applied  to  mathe- 


INDUCTION    IN    ARITHMETIC.  20 i 

rnatics,  and  not  a  few  of  them  distinctly  assert  it.  Dr. 
Whewell  says,  in  speaking  of  mathematics:  "These  sciences 
have  ....  no  process  of  proof  but  deduction."  Prof. 
Dodd  wrote  several  pamphlets  to  prove  that  there  can  be  no 
such  thing  as  inductive  reasoning  in  arithmetic ;  and  several  of 
those  whom  he  criticised  in  these  articles,  have  acknowledged 
the  correctness  of  his  views,  and  consequently,  their  own 
mistakes. 

These  views,  I  have  already  shown,  are  only  partially  true 
Arithmetic  is  a  deductive  science;  all  of  its  truths  may  prob- 
ably be  derived  by  deduction;  but  it  is  equally  true  that  some 
of  them  may  also  be  obtained  by  induction,  as  has  been  shown 
above  ;  and  also,  that  some  of  them  are  accepted  alone  on 
induction,  having  never  been  demonstrated. 

Great  care  should  be  exercised,  however,  in  the  use  of  induction 
in  mathematics.  Several  supposed  truths  which  were  derived 
by  induction  were  subsequently  found  to  be  untrue.  Fermat 
asserted  that  the  formula,  2m-f  1  is  always  a  prime,  when 
m  is  taken  any  term  in  the  series  1,  2,  4,  8,  16,  etc.,  but 
Euler  found  that  232-f- 1  is  a  composite  number.  Lagrange 
tells  us  that  Euler  found  by  induction  the  following  rule  for 
determining  the  resolvability  of  every  equation  of  the  form 
xi-\-Ay=B,  when  B  is  a  prime  number:  the  equation  must  be 
possible  when  B  shall  have  the  form,  4A?i+r2,  or  4An+r2-  A. 
This  proposition  holds  good  for  a  large  number  of  cases,  and 
was  thought  by  many  mathematicians  to  be  entirely  general, 
but  the  equation,  x1—  79y2=101,  Lagrange  proves  to  be  an 
exception  to  it. 

The  danger  of  inductive  inference  in  mathematics  is  also  seen 
in  some  of  the  formulas  which  have  been  presented  for  finding 
prime  numbers.  Several  of  these  hold  good  for  many  terms, 
and  were  supposed  to  be  general,  but  were  at  last  found  to  be 
only  special.  Thus,  the  formula  x2  -J-#-j-41  holds  good  for  forty 
values  of  x.  The  formula  x2+ x-\-  17  gives  seventeen  of  its  first 
values  prime,  and  2j;2+29  gives  twenty-nine  of  its  first  values 
prime. 


WA  THE    PHILOSOPHY    OF    ARITHMETIC. 

Having  shown  that  mathematics,  though  a  deductive  sci- 
ence, will  admit,  in  some  instances,  of  an  inductive  treatment, 
it  may  be  remarked  that  such  treatment  is  especially  adapted  to 
young  pupils  in  the  elementary  processes  of  arithmetic.  It  is 
difficult  for  them  to  draw  conclusions  from  the  principles  estab- 
lished by  a  deductive  demonstration ;  hence,  in  some  cases,  it 
may  be  well  for  them  to  employ  the  inductive  method.  The 
rules  for  working  fractions  may  be  derived  by  an  inductive 
inference  from  the  solution  of  a  particular  example;  and  this 
method  will  be  much  more  readily  understood  than  the  deriva- 
tion of  them  from  general  principles  deductively  established. 
The  method  is  to  solve  a  particular  problem  by  analysis,  and 
then  derive  a  general  method  by  an  inductive  inference  from 
such  analysis.  Thus  analysis  and  induction  become,  as  it 
were,  golden  keys  with  which  we  unlock  the  complex  combina- 
tions of  numbers. 

It  will  be  well,  however,  to  lead  the  pupils  to  the  deductive 
method  as  soon  as  possible.  Most  students  will  make  the 
transition  naturally.  The  better  reasoners  among  them  will 
themselves  rise  from  this  inductive  method,  being  satisfied 
only  with  a  deductive  demonstration;  and  in  this  they  should 
be  encouraged.  They  will  often  see  the  deductive,  or  necessary 
idea,  behind  the  inductive  process,  and  thus  pass  spontaneously 
from  the  particular  fact  to  the  general  truth.  They  will  some- 
times discover  a  truth  by  trial  and  inference,  that  is,  by  induc- 
tion, and  then  learn  to  demonstrate  it  deductively ;  and  it  will 
be  a  useful  exercise  for  pupils  to  have  some  special  drill  in  this 
manner.  They  will  thus  see  the  relation  of  the  two  methods 
of  reasoning,  and  be  impressed  with  the  deductive  nature  of 
the  science  of  arithmetic,  and  the  necessary  character  of  its 
truths. 


Part  1 1. 

SYNTHESIS  AND  ANALYSIS. 


SECTION    I. 
FUNDAMENTAL  OPERATIONS 


SECTION    II. 
DERIVATIVE  OPERATIONS 


SECTION    i. 

FUNDAMENTAL  OPERATIONS  OF   SYNTHESIS 
AND  ANALYSIS 


I.    Addition. 


II.     Subtraction. 


III.     Multiplication 


IV.     Division. 


CHAPTER  I. 

ADDITION. 

THE  fundamental  synthetic  process  of  arithmetic  is  Addi- 
tion. Beginning  at  the  Unit  as  the  primary  numerical 
idea,  numbers  arise  by  a  process  of  synthesis.  By  it  we  pass 
from  unity  to  plurality;  from  the  one  to  the  many.  This 
mental  process  which  gives  rise  to  numbers,  we  naturally 
extend  to  the  numbers  themselves,  and  thus  synthesis  becomes 
the  primary  operation  of  arithmetic.  This  general  synthetic 
process  is  called  Addition. 

Definition. — Addition  is  the  process  of  finding  the  sum  of 
two  or  more  numbers.  The  sum  of  two  or  more  numbers  is 
a  single  number  which  expresses  as  many  units  as  the  several 
numbers  added.     The  sum  is  often  called  the  amount. 

Addition  may  also  be  denned  as  the  process  of  uniting  sev- 
eral numbers  into  one  number  which  expresses  as  many  units 
as  the  several  numbers  united.  This  last  definition  includes 
both  of  the  previous  ones,  and  avoids  the  use  of  the  word  sum. 
The  former  definition  is,  however,  preferred  on  account  of 
its  conciseness  and  simplicity,  and  is  the  one  usually  adopted 
by  arithmeticians. 

Principles.  —  The  process  of  addition  is  performed  in 
accordance  with  certain  necessary  laws  which  are  called  prin- 
ciples.    The  most  important  of  these  are  the  following: 

I.  Only  similar  numbers  can  be  added.  Thus,  we  cannot 
find  the  sum  of  4  apples  and  5  peaches,  for  if  we  unite  the 
numbers  we  shall  have  neither  9  apples  nor  9  peaches.     It  haa 

i  207  ) 


208  THE    PHILOSOPHY    OF    ARITHMETIC. 

been  claimed,  that  the  sum  is  9  apples  and  peaches ;  in  proof 
of  which  it  is  said  we  speak  properly  of  "  12  knives  and  forks." 
meaning  6  knives  and  6  forks.  Such  a  combination  is,  how- 
ever, popular  rather  than  scientific;  it  is  not  what  we  mean  by 
a  strict  use  of  the  word  addition. 

It  may  also  be  observed  that  dissimilar  numbers  may  be 
brought  under  the  same  name  and  thus  become  similar,  when 
they  can  be  united  in  one  sum.  Thus,  4  sticks  and  5  stones 
may  be  regarded  as  so  many  objects  or  things,  and  their  sum 
will  be  9  objects  or  9  things.  So  in  writing  units  and  tens  in 
the  Arabic  system;  they  cannot  be  combined  directly,  but  by 
reducing  both  to  tens  or  both  to  units,  the  addition  can  be 
effected. 

II.  The  sum  is  a  number  similar  to  the  numbers  added. 
This  is  evidently  an  axiomatic  truth.  The  sum  of  4  cows  and 
5  cows  is  9  cows,  and  cannot  be  horses  or  sheep,  or  anything 
besides  cows.  An  apparent  exception  which  will  be  under- 
stood by  what  is  said  above  is,  that  the  sum  of  3  horses  and 
5  cows  is  8  animals. 

III.  The  sum  is  the  same  in  whatever  order  the  numbers 
are  added.  This  is  evident  from  the  consideration  that  in  any 
case  we  have  the  combination  of  the  same  number  of  units, 
and  consequently  the  same  sum. 

Cases. — Addition  is  divided  philosophically  into  two  gen- 
eral cases.  The  first  case  consists  in  finding  the  sums  of 
numbers  independently  of  the  notation  used  to  express  them. 
The  second  case  consists  in  finding  the  sum  of  numbers  as 
expressed  in  written  characters,  and  thus  grows  out  of  the  use 
of  the  Arabic  system  of  notation.  The  former  deals  with  small 
[lumbers  which  can  be  united  mentally,  and  may  be  called 
mental  addition  ;  the  latter  is  used  with  large  numbers  as 
expressed  with  written  characters,  and  may  be  called  written 
addition.  The  former  is  a  process  of  pure  arithmetic;  tluj 
latter  is  incidental  to  the  system  of  notation  which  may  be 
employed,  and  is  not  essential  to  number  in   itself  considered 


ADDITION.  209 

The  former  method  is  an  independent  process,  complete  in 
itself;  the  latter  is  dependent  upon  the  former  for  the  elements 
with  which  it  works.  By  the  former  case  we  obtain  what  we 
may  call  the  primary  sums  of  addition,  or  what  is  generally 
known  as  the  Addition  Table,  which  we  make  use  of  in  adding 
large  numbers  expressed  by  the  Arabic  method  of  notation. 

Treatment. — The  primary  synthetic  arithmetical  process  is 
that  of  increasing  by  units.  This  process  is  presented  in  the 
genesis  of  numbers  where,  by  counting,  we  pass  from  one  num- 
ber to  another  immediately  following  it,  by  the  addition  of  a 
unit;  and  it  also  lies  at  the  foundation  of  the  method  by  which 
we  find  the  sum  of  any  two  or  more  numbers.  By  it  we  obtain 
the  elementary  sums  of  the  first  case,  and  then  we  use  these 
sums  in  solving  the  problems  of  the  second  case.  The  method 
of  treating  both  of  these  cases  will  be  presented  somewhat  in 
detail. 

Case  I.  To  find  the  primary  sums  of  arithmetic. — The 
primary  sums  of  arithmetic  are  found  by  the  same  process  of 
counting  by  which  our  ideas  of  numbers  are  generated.  The 
sum  of  two  numbers  is  primarily  determined  by  beginning  at 
one  number  and  counting  forward  from  it  as  many  units  as  are 
in  the  number  to  be  added  to  it.  Thus,  to  find  the  sum  of  any 
two  numbers,  &s  five  and  four,  we  begin  at  five  and  count  four 
successive  numbers,  six,  seven,  eight,  nine,  and  seeing  we 
reach  nine,  we  know  th&t  five  and  four  are  nine.  In  this  way 
we  obtain  the  sums  of  all  small  numbers,  and  then  commit 
them  to  memory,  that  we  may  know  them  when  we  wish  to 
use  them  without  passing  through  the  steps  by  which  they 
were  obtained. 

To  be  assured  that  this  is  the  real  method,  we  have  but  to 
watch  young  children  when  adding,  and  we  shall  see  that  they 
do  actually  find  the  sums  of  numbers  in  the  manner  explained. 
They  may  often  be  seen  counting  their  fingers,  or  marks  on  the 
slate,  in  performing  addition.  The  elementary  sums  thus 
found  are  the  basis  of  addition.  We  fix  them  in  the  memory 
14 


210  THE    PHILOSOPHY    OF    ARITHMETIC. 

as  we  do  the  elementary  products  of  the  multiplication  table, 
and  employ  them  in  finding  the  sums  of  larger  numbers. 

These  primary  sums  may  be  regarded  as  the  axioms  of 
addition.  They  are  intuitive  truths,  that  is,  truths  which  can- 
not be  demonstrated,  but  are  seen  by  intuition.  "Why  is  it," 
says  Whewell,  "that  three  and  two  are  equal  to  four  and  one? 
Because  if  we  look  at  five  things  of  any  kind,  we  see  that  it  is 
so.  The  five  are  four  and  one;  they  are  also  three  and  two 
The  truth  of  our  assertion  is  involved  in  our  being  able  to 
conceive  the  number  five  at  all.  We  perceive  this  truth  by 
intuition,  for  we  cannot  see,  or  imagine  we  see,  five  things, 
without  perceiving  also  that  the  assertion  above  stated  is 
true." 

Case  II.  To  add  numbers  expressed  by  the  Arabic  system 
of  notation. — The  principle  by  which  we  find  the  sum  of 
larger  numbers  expressed  by  the  Arabic  system,  is  that  of 
adding  by  parts.  Having  learned  the  sums  of  small  numbers, 
we  separate  larger  numbers  into  parts  corresponding  to  these 
small  numbers,  and  then  find  the  sum  of  these  parts  which, 
united,  will  give  the  entire  sum.  Thus  in  practice  we  first  add 
the  units  group,  then  the  tens  group,  and  thus  continue  until 
all  the  groups  are  added.  If  the  sum  of  any  group  amounts 
to  more  than  nine  units  of  that  group,  we  incorporate  the  tens 
term  of  the  sum  with  the  sum  of  the  next  higher  group. 

Solution. — Thus,  in  adding  the  two  numbers  368  and  579, 

we  write  the  numbers   so  that  similar  terms 

stand  in  the   same   column,  and  begin   at  the     operation. 

right  to  add.    9  units  and  8  units  are  IT  units,  368 

or  1  ten  and  7  units;  we  write  the  7  units,  and 

947 
add  the  1  ten  to  the  sum  of  the  next  column. 

7  tens  and  6  tens  are  13  tens,  and  1  ten  are  14  tens,  or  1  hundred 

and  4  tens ;  we  write  the  4  tens,  and  add  the  1  hundred  to  the 

next  column.     5  hundreds  and  3  hundreds  are  8  hundreds,  and 

1  hundred  are  9  hundreds,  which  we  write  in  hundreds  place 

The  entire  sum  is  therefore  947. 


ADDITION.  211 

This  method  of  adding  by  parts  is  the  result  of  the  beautiful 
system  of  Arabic  notation,  whereby  figures  in  different  positions 
express  groups  of  different  value.  It  is  peculiar  to  this  method 
of  expressing  numbers,  and  illustrates  its  great  convenience  and 
utility.  In  adding  large  numbers,  it  would  be  exceedingly  dif- 
ficult, if  not  impossible,  for  the  mind  to  unite  them  directly  into 
one  sum ;  but  by  adding  the  groups  separately,  the  process  is 
simple  and  easy. 

Rule. — One  of  the  most  common  errors  of  arithmetic  is  found 
in  the  statement  of  the  rules  of  the  fundamental  operations. 
This  error  consists  in  confounding  the  meaning  of  the  words 
figure  and  number.  Thus,  it  is  usual  to  speak  of  "adding  the 
figures,"  of  "carrying  the  left-hand  figure  to  the  next  column,'* 
etc.  This  is  a  mistake  involving  a  looseness  of  thought  that 
ought  not  to  be  permitted  to  remain  in  the  text-books.  We 
cannot  add  figures,  we  can  add  only  the  numbers  which  they 
express. 

This  error  can  be  avoided  in  several  ways.  The  method 
here  suggested  is  the  use  of  the  word  term  for  figure.  The 
word  term  is  already  employed  in  a  similar  manner  in  algebra. 
It  may  be  used  in  a  dual  sense,  embracing  both  the  figure  and 
the  number  expressed  by  the  figure.  Numbers  and  figures 
have  a  definite  signification,  and  one  cannot  be  used  for  the 
other  without  a  mistake ;  but  it  will  be  both  correct  and  con- 
venient to  use  one  word  for  both.  No  ambiguity  will  be  occa- 
sioned by  it,  as  the  particular  meaning  may  be  determined  by 
the  application.  In  this  way  we  may  avoid  the  error  of  speak- 
ing of  "adding  figures,"  and  also  the  inconvenient  expression 
sometimes  employed  of  "adding  the  numbers  denoted  by  the 
figures." 

Why  do  we  write  the  numbers  as  suggested,  and  why  do 
we  begin  at  the  right  hand  to  add,  are  questions  very  fre- 
quently asked  of  the  arithmetician.  In  adding  numbers  we 
write  them  one  under  another,  so  that  figures  of  the  same 
order  stand  in  the   same  vertical  column,  for  convenience   in 


212  THE    PHILOSOPHY    OF    ARITHMETIC. 

adding.  We  begin  at  the  right  hand  to  add  as  a  matter  of 
convenience  also,  so  that  when  the  sura  of  any  column  exceeds 
nine  units  of  that  column,  we  may  unite  the  number  denoted 
by  the  left  hand  term  to  the  next  column.  We  can  also  add  by 
beginning  at  the  left,  but  it  will  be  seen  on  trial  to  be  much  less 
convenient.  We  commence  at  the  bottom  of  a  column  to  add 
as  a  matter  of  custom ;  in  practice  it  is  sometimes  more  con- 
venient to  begin  at  the  bottom  and  at  other  times  at  the  top. 

Were  the  scale  any  other  than  the  decimal,  the  principle  and 
method  of  adding  would  be  the  same.  In  addition  of  denom- 
inate numbers,  where  the  scales  are  irregular,  the  same  general 
principle  is  employed.  We  find  the  sum  of  a  lower  order  of 
units,  reduce  this  to  the  next  higher  order,  etc.  The  difference 
in  practice  is  that,  with  the  decimal  scale,  the  reduction  is  evi- 
dent from  the  notation,  while  in  the  irregular  scales  we  must 
divide  to  make  the  reduction.  The  general  principle  of 
thought  in  the  two  cases  is,  however,  identical. 


CHAPTER  II. 

SUBTRACTION. 

rpHE  fundamental  analytical  process  of  arithmetic  is  Sub- 
i.  traction.  This  process  arises  from  the  reversing  of  the 
fundamental  synthetic  process.  The  primary  operation  of 
arithmetic,  as  previously  seen,  is  synthesis.  Every  synthesis 
implies  a  corresponding  analysis;  hence,  the  second  operation 
of  arithmetic,  as  a  logical  consequence,  must  be  the  oppo- 
site of  the  primary  synthetic  process.  In  the  former  case  we 
united  numbers  to  find  a  sum ;  here  we  separate  numbers  to 
find  a  difference.  This  general  analytic  process  has  received 
the  name  of  Subtraction. 

Definition. — Subtraction  is  the  process  of  finding  the  differ- 
ence between  two  numbers.  The  difference  between  two  num- 
bers is  a  number  which  added  to  the  less  will  give  a  sum  equal 
to  the  greater.  The  greater  number  is  called  the  Minuend;  the 
less  number  is  called  the  Subtrahend.  Subtraction  may  also 
be  defined  as  the  process  of  finding  how  much  greater  one 
number  is  than  another;  or,  as  the  process  of  finding  a  num- 
ber which,  added  to  the  smaller  of  two  numbers,  will  equal 
the  greater.  The  definition  first  presented  is,  however,  pre- 
ferred. 

Cases. — Subtraction  is  philosophically  divided  into  two 
general  cases,  like  addition.  The  first  case  consists  in  finding 
the  difference  between  two  numbers,  independent  of  the  nota- 
tion used  to  express  them.  The  second  case  consists  in  find- 
ing the  difference  between  numbers  as  expressed  in  written 

(213) 


214  THE    PHILOSOPHY   OF    ARITHMETIC. 

characters,  and  thus  grows  out  of  the  use  of  the  Arabic  nota 
don.  The  first  is  a  case  of  pure  arithmetic,  independent  of  any 
notation  ;  the  latter  is  incidental  to  the  notation  adopted  to 
express  numbers.  The  former  deals  with  small  numbers,  and 
the  process  being  wholly  in  the  mind  may  be  called  Mental 
Subtraction ;  the  latter  is  employed  in  subtracting  large  num- 
bers expressed  with  written  characters,  and  may  be  called 
Written  Subtraction.  The  former  is  an  independent  process 
complete  in  itself;  the  latter  has  its  origin  in  the  Arabic  system 
of  notation,  and  is  dependent  upon  the  former  for  its  elementary 
differences.  In  the  ordinary  text-books,  the  second  case  is 
usually  divided  into  two  separate  cases,  depending  upon  the 
size  of  the  terms  in  the  minuend  and  subtrahend;  but  such 
division  is  designed  to  simplify  the  subject  in  instruction,  and 
is,  therefore,  a  practical  rather  than  a  logical  division  of  the 
subject. 

Principles. — The  operations  in  subtraction  depend  upon  some 
general  laws  called  principles.  The  most  important  of  the  fun- 
lamental  principles  of  subtraction  are  the  following: 

1.  Similar  numbers  only  can  be  subtracted.  Thus,  we  can- 
lot  find  the  difference  between  9  apples  and  4  peaches,  for  if 
vve  take  the  difference  between  the  numbers  9  and  4,  which  is 
5,  it  will  be  neither  5  apples  nor  5  peaches.  Suppose,  how- 
ever, that  we  have  9  apples  and  peaches,  consisting  of  5  apples 
and  4  peaches;  can  we  then  subtract  4  peaches,  and  will  not  the 
remainder  be  5  apples?  Or  suppose  we  have  a  collection  of 
knives  and  forks  consisting  of  half  a  dozen  of  each,  which  are 
sometimes  spoken  of  as  "  12  knives  and  forks;"  can  we  noi 
take  away  6  forks  and  leave  remaining  6  knives  ?  In  reply,  we 
remark  that  such  a  "taking  away"  is  not  what  we  mean  by 
subtraction,  which  is  defined  as  the  process  of  finding  the 
difference  of  two  numbers. 

It  is  also  manifest,  as  in  addition,  that  if  we  regard  the  dis- 
similar numbers  as  having  the  same  generic  name,  they 
will  then  become  similar  and  we  can  subtract  them.     Thus,  9 


SUBTRACTION.  215 

apples  and  4  peaches  may  be  regarded  as  9  objects  and  4 
objects,  the  difference  of  which  is  5  objects.  So  in  subtracting 
the  different  orders  of  units  in  the  Arabic  scale,  we  cannot  sub- 
tract them  directly  as  different  orders,  but  by  reducing  them 
to  the  same  denomination,  the  subtraction  is  readily  performed. 

2.  The  difference  is  a  number  similar  to  the  minuend  and 
subtrahend.  This  is  a  necessary  truth  intuitively  apprehended. 
Thus  4  men  subtracted  from  9  men,  leaves  5  men,  and  not 
5  girls,  or  5  women.  If  we  have  a  group  consisting  of  9 
persons,  5  men  and  4  women,  and  take  away  4  women,  there 
will  remain  5  men  ;  hence  we  might  infer  that  4  women  taken 
from  9  persons  leaves  5  men  ;  but  this  is  not  a  universal  truth; 
neither,  as  stated  above,  is  such  a  taking  away,  what  we  mean 
by  subtraction. 

3.  If  the  minuend  and  subtrahend  be  equally  increased 
or  diminished,  the  remainder  will  be  the  same.  This  is  in- 
cluded in  the  axiom  that  the  difference  between  two  numbers 
equals  the  difference  between  them  when  equally  increased  or 
diminished.  The  truth  of  such  a  proposition  is  seen  to  be 
necessary  as  soon  as  the  proposition  is  clearly  apprehended  by 
the  mind. 

4.  The  minuend  equals  the  sum  of  the  subtrahend  and 
remainder;  the  subtrahend  equals  the  difference  between  the 
minuend  and  remainder.  These  two  principles  flow  from  the 
conception  of  subtraction,  and  the  relation  of  the  several  terms 
to  one  another.  Given  a  clear  idea  of  the  process  of  subtraction, 
and  the  relation  of  the  three  terms  in  the  process,  and  these 
truths  immediately  follow. 

Method.  —  The  two  cases  of  subtraction,  as  of  addition, 
require  distinct  methods  of  treatment.  In  the  former  case  we 
subtract  directly  as  wholes,  finding  the  difference  by  reversing 
the  process  of  addition.  In  the  latter  case  we  subtract  by 
parts,  using  the  elementary  differences  to  find  the  differences 
of  the  corresponding  parts.  An  explanation  of  both  cases  wil* 
be  presented. 


216  THE    PHILOSOPHY    OF    ARITHMETIC. 

Case  I.  To  find  the  primary  differences  in  arithmetic. — 
The  elementary  differences  are  obtained  by  a  reversion  of 
the  process  of  finding  the  elementary  sums.  This  may  be 
done  in  two  distinct  ways.  First,  we  may  find  the  difference 
between  two  numbers  by  counting  off  from  the  larger  number 
as  many  units  as  are  contained  in  the  smaller  number.  Thus, 
if  we  wish  to  subtract  four  from  nine,  we  may  begin  at  nine 
and  count  backward  four  units:  thus,  eight,  seven,  six,  five ; 
and  finding  that  we  reach  five,  we  know  that  four  from  nine 
leaves  five.  This  is  the  reverse  of  the  process  by  which  we 
obtained  the  elementary  sums  in  addition.  In  one  case  we 
count  on  for  the  sum ;  in  the  other  we  count  off  for  the  differ- 
ence. 

The  other  method  consists  in  finding  the  elementary  differ- 
ences by  deriving  them  by  inference  from  the  elementary 
sums.  Thus,  in  finding  the  difference  between  five  and  nine, 
we  may  proceed  as  follows:  since  four  added  to  five  equals 
nine,  nine  diminished  by  five,  equals  four.  This  process,  put 
in  a  formal  manner,  is  as  follows :  The  difference  between  two 
numbers  is  a  number  which,  added  to  the  less,  will  equal  the 
greater;  but,  four  added  to  five,  the  less,  equals  nine,  the 
greater;  hence,  four  is  the  difference  between  nine  and  five. 
In  other  words,  we  know  that  five  from  nine  leaves  four, 
because  four  added  to  five  equals  nine. 

The  difference  between  these  two  methods  is  radical.  By 
the  former  method  we  derive  the  difference  by  direct  intuition, 
as  we  obtained  the  sums  in  addition.  We  see  that  the  differ- 
ence is  five.  By  the  second  method  we  infer  that  the  differ- 
ence \s  five,  without  directly  seeing  it.  The  latter  is  a  process 
of  reasoning,  and  will  admit  of  being  reduced  to  the  form  of 
a  syllogism,  as  is  shown  above.  The  point  made  here  is  an 
important  one,  and  will  throw  some  light  on  the  nature  of 
the  science  of  arithmetic,  which,  by  the  metaphysicians,  has 
been  somewhat  imperfectly  understood. 

The  second  method  is  preferred  in  practice  to  the  first,  as  we 


SUBTRACTION.  217 

can  make  use  of  the  elementary  sums  in  finding  the  elementary 
differences.  If  the  first  method  is  used,  it  will  be  necessary  to 
commit  the  elementary  differences  as  well  as  the  elementary 
sums.  By  making  the  differences  depend  upon  the  sums,  this 
labor  will  be  avoided. 

Case  II.  To  subtract  numbers  expressed  by  the  Arabic 
scale  of  notation.  With  large  numbers  we  cannot  subtract  the 
one  directly  from  the  other  as  with  small  numbers ;  we  there- 
fore divide  the  labor,  subtracting  by  parts;  that  is,  we  find 
the  difference  between  the  corresponding  groups  of  each  term. 
By  this  means  the  labor  of  subtracting  is  greatly  facilitated,  so 
that  with  large  numbers,  which  it  would  be  almost,  if  not 
quite  impossible  otherwise  to  subtract,  the  operation  becomes 
simple  and  easy. 

In  the  subtraction  of  numbers  expressed  in  the  Arabic  scale 
of  notation,  two  distinct  cases  arise;  first,  when  the  number 
of  each  group  of  the  subtrahend  does  not  exceed  the  correspond- 
ing number  of  the  minuend ;  second,  when  the  number  of  a 
group  in  the  subtrahend  exceeds  the  corresponding  number  in 
the  minuend.  In  the  first  case  we  readily  subtract  each  group 
in  the  subtrahend  from  the  corresponding  group  in  the  minu- 
end. In  the  second  case  a  difficulty  arises,  for  which  we  have 
two  distinct  methods  of  explanation,  called  respectively  the 
Method  by  Borrowing,  and  the  Method  by  Adding  Ten. 

To  illustrate  these  methods,  suppose  it  be  required  to  sub 
tract  526  from  874. 

First  Method. — Having  the   numbers  writ-     operation. 
ten  as  in  the   margin,  we    commence  at  the  8?4 

right  to  subtract,  and  reason  thus:  we  cannot 

take  6  units  from  4  units,  we  will   therefore  **4° 

take  1  ten  from  the  7  tens,  and  add  it  to  the  four  units,  which 
will  give  14  units.  We  then  subtract  6  units  from  14  units, 
which  gives  8  units.  We  then  subtract  2  tens  from  the  6  tens 
which  remain  after  taking  away  the  1  ten,  which  leaves  4. 
tens.  We  also  subtract  5  hundreds  from  8  hundreds,  leaving 
3  hundreds:  hence  the  difference  is  348. 


2.6  THE    PHILOSOPHY    OF    ARITHMETIC. 

Second  Method. — By  the  second  method  we  reason  thus: 
We  cannot  subtract  6  units  from  4  units,  hence  we  add  10  to 
the  4,  making  14  units,  and  then  say,  6  units  from  14  units 
leave  8  units.  Now,  since  we  have  added  10  to  the  minuend, 
that  the  remainder  may  be  correct  we  must  add  one  ten  to  the 
subtrahend ;  hence  we  have  3  tens  from  7  tens  leave  4  tens, 
and  also  as  before,  5  hundreds  from  8  hundreds,  3  hundreds 
This  solution  is  founded  upon  the  principle  that  the  difference 
between  two  numbers  equals  the  difference  between  the  two 
numbers  equally  increased. 

The  first  method  seems  preferable  on  account  of  its  simpli- 
city of  thought,  as  it  merely  changes  the  form  of  the  minuend. 
Pupils  see  the  reason  of  the  process  by  this  method  more 
readily  than  by  the  method  of  adding  ten.  The  second  method, 
however,  is  preferred  by  some  teachers  for  at  least  two  reasons. 
First,  it  is  the  method  generally  used  in  practice;  nearly  all 
persons  increasing  the  next  lower  term  after  "  borrowing," 
instead  of  diminishing  the  upper  one.  Second,  it  is,  in  many 
cases  which  arise,  much  more  convenient  than  the  other 
method,  as  in  subtracting  12345  from  20000.  By  the  second 
method,  the  solution  of  this  problem  will  be  much  simpler 
than  by  the  first. 

Another  Method. — There  is  still  another  method  of  subtract- 
ing, which,  if  not  of  any  practical  value,  is  at  least 
of  sufficient  interest  to  be  worthy  of  mention.     It       74682 

O  H  o  p  r 

consists  in  subtracting  the  terms  of  the  subtrahend 
from  10,  and  adding  the  difference  to  the  corres-  46817 
ponding  terms  of  the  minuend.  Thus,  in  subtracting  27865 
from  74682,  we  say  5  from  10  leaves  5,  and  2  are  7 ;  6  and  1  to 
carry  are  7,  and  7  from  10  leaves  3,  and  8  are  11 ;  set  down  the 
1;  8  from  10  leaves  2,  and  6  are  8;  7  and  1  to  carry  are  8,  and 
8  from  10  leaves  2,  and  4  are  6,  etc. 

Rule. — In  the  rule  for  subtraction,  arithmeticians  make  the 
«ame  mistake  as  in  the  rule  for  addition.  Thus,  they  say, 
"  Subtract  each  figure  of  the  subtrahend  from  the  figure  abo^e 


SUBTRACTION.  219 

it  in  the  minuend,"  or  "  take  each  figure  of  the  subtrahend 
from  the  figure  above  it,"  or,  "if  a  figure  in  the  lower  number 
is  larger  than  the  one  above  it,"  etc.  These  errors  are  almost 
inexcusable.  We  cannot  subtract  figures,  we  subtract  num- 
bers. If  we  "take  one  figure  from  another"  the  other  figure 
will  be  left,  not  the  difference  of  the  numbers  expressed  by 
them.  A  figure  is  larger  or  smaller  according  to  the  kind  of 
type  in  which  it  is  printed.  The  figure  two  may  be  large  (2) 
or  small  (2).  One  figure  may  be  larger  than  another,  and 
express  a  smaller  number;  as,  3  and  8. 

This  error  may  be  avoided  by  the  use  of  the  word  term  for 
the  number  expressed  by  the  figure.  The  rule  will  then  read, 
"Begin  at  the  right  and  take  each  term  of  the  subtrahend  from 
the  corresponding  term  of  the  minuend,"  etc.  "If  a  term  of 
the  subtrahend  is  greater  than  the  corresponding  term  of  the 
minuend,"  etc. 

Remarks. — We  write  terms  of  the  same  order  in  the  same 
vertical  column  for  convenience  in  subtracting,  since  only  num- 
bers of  the  same  group  can  be  subtracted.  We  commence  at 
the  right,  so  that  when  a  term  of  the  subtrahend  expresses 
more  units  than  the  corresponding  term  of  the  minuend,  we 
may  take  it  from  the  next  higher  group  of  the  minuend  ;  or,  if 
we  use  the  other  method  of  subtracting,  that  we  may  add  10 
of  a  group  to  the  minuend,  and  1  of  the  next  higher  group  to 
the  subtrahend;  in  other  words,  we  commence  at  the  right  as 
a  matter  of  convenience,  as  will  be  seen  in  the  attempt  to  sub- 
tract by  commencing  at  the  left. 

The  taking  one  from  the  next  term  of  the  minuend  is  called 
"borrowing,"  and  the  adding  one  to  the  next  term  of  the  sub- 
trahend is  called  "carrying."  The  accuracy  of  these  words 
has  been  questioned.  To  borrow  is  to  obtain  that  which  we 
expect  to  return  to  the  one  from  whom  we  borrow.  It  does 
not  seem  much  like  "  borrowing"  to  take  from  one  thing  and 
return  what  we  take  to  another.  It  is  something  like  "robbing 
Peter  to  pay  Paul."      In  regard  to  the  term  "carrying,"  it 


220  THE    PHILOSOPHY    OF    ARITHMETIC. 

may  be  asked  in  what  it  is  carried ;  though  we  may  answer, 
as  the  boy  did,  "we  carry  in  the  head."  Notwithstanding 
these  objections,  the  terms  borrowing  and  carrying  have  been 
sanctioned  by  good  usage ;  and,  since  custom  is  the  lawgiver 
in  language,  we  may  accept  them  as  correct.  Their  use  is  a 
matter  of  convenience,  also,  as  they  indicate  operations  for 
which  we  have  no  other  technical  terms.  It  may  be  remarked 
that  it  required  many  years  for  the  people  of  Europe  to  become 
familiar  with  the  processes  of  borrowing  and  carrying.  In  a 
work  on  arithmetic  by  Bernard  Lamy,  published  at  Amster- 
dam in  1692,  the  author  states  that  a  friend  sends  him  the 
mode  of  using  the  carriage  in  subtraction,  he  having  previ- 
ously borrowed  from  the  upper  line ;  and  this  is  presented  as 
a  novelty. 


CHAPTER  III. 

MULTIPLICATION. 

THE  general  process  of  synthesis  is  Addition.  Having 
become  familiar  with  this  general  synthetic  process  in  ac- 
cordance with  the  law  of  thought,  from  the  universal  to  the 
particular,  we  begin  to  impose  certain  conditions  upon  it. 
The  numbers  primarily  united  were  of  any  relative  value;  if, 
now,  we  impose  the  condition  that  the  numbers  united  shall  be 
all  equal,  with  the  new  idea  of  the  times  the  number  is  used, 
we  have  a  new  process  of  synthesis,  which  we  call  Multiplica- 
tion. 

Multiplication  is  thus  seen  to  be  a  special  case  of  addition, 
in  which  the  numbers  added  are  all  equal.  The  idea  of  mul- 
tiplication is  contained  in  addition,  and  is  an  outgrowth  of  it. 
They  are  both  synthetic  processes— one  being  a  general,  and 
the  other  a  more  special  synthesis.  Multiplication,  however, 
involves  the  idea  of  "  times,"  which  does  not  appear  in  addi- 
tion. This  notion  of  "  times,"  originating  in  multiplication,  is 
one  of  the  most  important  in  mathematics,  and  is  itself  the 
source  of  a  large  portion  of  the  science.  Thus,  in  involution 
there  is  no  apparent  trace  of  the  idea  of  addition,  and  the  same 
is  true  in  respect  of  other  processes.  If,  however,  we  follow 
these  processes  back  far  enough,  we  shall  find  they  have  their 
origin  in  the  primary  process  of  addition.  Even  involution 
may  be  performed  by  successive  additions. 

Definition.— Multiplication   is  the    process  of   finding  the 
product  of  two  numbers.     The   Product  of  two  numbers  is 

(221) 


222  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  result  obtained  by  taking  one  number  as  many  times  as 
there  are  units  in  the  other.  The  number  multiplied  is  called 
the  Multiplicand.  The  number  by  which  we  multiply  is  called 
the  Multiplier. 

This  definition  of  multiplication,  introducing  the  word  Pro- 
duct, makes  it  similar  to  the  definitions  of  addition  and  sub- 
traction, in  which  the  terms  sum  and  difference  are  used. 
Defining  Division  in  a  similar  manner  by  using  the  word  Quo- 
tient, we  shall  have  a  harmony  in  the  definitions  of  the  four 
fundamental  rules,  which  has  not  hitherto  existed.  I  have 
adopted  this  method  in  my  Higher  Arithmetic,  and  shall  intro- 
duce it  into  my  other  mathematical  works. 

Multiplication  is  usually  defined  as  the  process  of  taking  one 
number  as  many  times  as  there  are  units  in  another.  This 
definition  is  not  entirely  satisfactory.  It  says  nothing  about 
finding  a  result,  which  is  specified  in  the  definitions  of  addition 
and  subtraction,  and  which  seems  to  be  necessary  also  here. 
To  supply  this  omission,  I  have  previously  defined  multiplica- 
tion as  the  process  of  finding  the  result  of  taking  one  number 
as  many  times  as  there  are  units  in  another.  After  a  very 
careful  consideration  of  the  subject,  however,  I  have  concluded 
to  adopt  the  method  of  defining  multiplication  as  the  process  of 
finding  the  product,  thus  securing  a  uniformity  in  the  defini- 
tions of  the  fundamental  operations. 

Principles. — The  operations  of  multiplication  are  founded 
upon  certain  necessary  truths  called  principles.  The  most 
important  of  the  principles  of  multiplication  are  those  which 
follow : 

1.  The  multiplier  is  always  an  abstract  number.  For,  the 
multiplier  shows  the  number  of  times  the  multiplicand  is 
taken,  and  hence  must  be  abstract,  since  we  cannot  take  any- 
thing yards  times  or  bushels  times,  etc.  From  this  it  follows 
that  such  problems  as  "Multiply  25  cts.  by  25  cts.,"  or  "2s.  6d. 
by  itself"  are  impossible  and  absurd.  In  finding  areas  and 
volumes,  we  speak  of  multiplying  feet  by  feet  for  square  feet, 


MULTIPLICATION.  223 

square  feet  by  feet  for  cubic  feet,  etc.  It  should  be  remem- 
bered, however,  that  this  is  merely  a  convenient  expression, 
which  does  not  indicate  the  actual  process.  In  finding  the 
area  of  a  rectangle,  we  multiply  the  number  of  square  feet  on 
the  base  by  the  number  of  such  rows;  the  multiplicand  being 
square  feet  and  the  multiplier  an  abstract  number. 

2.  The  product  is  always  similar  to  the  multiplicand.  This 
is  manifest  from  the  fact  that  the  product  is  merely  the  sum  of 
the  multiplicand  used  as  many  times  as  there  are  units  in  the 
multiplier.  Thus,  3  times  4  apples  are  12  apples,  and  cannot 
be  12  pears  or  peaches. 

3.  The  product  of  two  numbers  is  the  same,  whichever  is 
made  the  multiplier.    This  may  be  seen  by  placing     *  *  *  * 

3  rows  of  4  stars  each  in  the  form  of  a  rectangle,     #  *  *  * 
as  in  the  margin.     Now  these  may  be  regarded     *  *  *  * 
as  3  rows  of  4  stars  each,  or  4  rows  of  3  stars 
each ;  hence  3  times  4  is  the  same  as  4  times  3 ;  and  the  same 
may  be  shown  for  any  other  two  numbers. 

4.  If  the  multiplicand  be  multiplied  by  all  the  parts  of  the 
multiplier,  the  sum  of  all  the  partial  products  will  be  the  true 
product.  This  grows  out  of  the  general  principle  that  the 
whole  is  equal  to  the  combination  of  all  of  its  parts.  It  is 
applied  in  finding  the  product  of  two  numbers  expressed  by 
the  Arabic  system. 

5.  The  multiplicand  equals  the  quotient  of  the  product 
divided  by  the  multiplier ;  the  multiplier  equals  the  quotient  of 
the  product  divided  by  the  multiplicand.  These  two  principles 
are  manifest  to  the  mind  as  soon  as  it  attains  a  clear  idea  of 
the  processes  of  multiplication  and  division,  and  the  relation 
of  the  two  to  each  other. 

Cases. — Multiplication  is  philosophically  divided  into  two 
general  cases.  The  first  case  consists  in  finding  the  products 
of  numbers  independently  of  the  method  of  notation  used  to 
express  them.  The  second  case  is  that  which  grows  out  of  the 
use  of  the  Arabic  system  of  notation.     The  former  deals  with 


224  THE    PHILOSOPHY    OF    ARITHMETIC. 

small  numbers  mentally,  and  may  be  called  Mental  Multiplica- 
tion; the  latter  deals  with  large  numbers,  expressed  by  means 
of  written  characters,  and  may  be  called  Written  Multiplica- 
tion. The  former  is  an  independent  process  complete  in  itself, 
and  belongs  to  pure  number;  the  latter  has  its  origin  in  the 
Arabic  system,  and  is  dependent  upon  the  former  for  its  ele- 
mentary products. 

Method. — The  general  method  is  to  find  the  product  of  small 
numbers  by  addition,  and  then  use  these  in  the  multiplication 
of  large  numbers.  The  first  case  is  thus  made  to  depend  upon 
addition,  and  the  second  case  upon  the  first  case.  Both  cases 
will  be  formally  presented. 

Case  I.  To  find  the  elementary  products  of  arithmetic. 
The  first  object  in  multiplication  is  to  find  the  elementary  pro- 
ducts. By  the  elementary  products  are  meant  the  products  of 
small  numbers  which,  arranged  together,  constitute  what  is 
called  the  Multiplication  Table.  These  elementary  products 
are  derived  by  addition.  Thus,  we  ascertain  that  four  times 
five  are  twenty,  by  finding,  by  actual  addition,  that  the  sum 
of  four  fives  is  twenty.  In  this  manner  all  the  elementary 
products  of  the  table  were  originally  obtained.  This  table  is 
committed  to  memory  in  order  to  save  labor  and  facilitate  the 
process  of  calculation.  We  are  thus  able  to  tell  immediately 
the  product  of  two  small  numbers,  which  otherwise  we  should 
be  obliged  to  obtain  by  an  actual  addition. 

The  elementary  products  are  not  derived  by  intuition,  and 
are  therefore  not  axioms;  they  are  the  result  of  a  process  of 
reasoning.  Thus,  in  order  to  find  the  product  of  three  times 
four,  we  may  reason  as  follows:  Three  times  four  is  equal 
to  the  sum  of  three  fours;  but  the  sum  of  three  fours,  we 
find  by  addition,  is  twelve  ;  hence,  three  times  four  is  twelve. 
This  is  as  valid  a  syllogism  as  "A  is  equal  to  B;  but  B  is 
equal  to  C;  hence,  A  is  equal  to  C." 

The  extent  of  the  table,  for  all  practical  purposes,  is  limited 
by  "nine  times  nine."     That  is,  with   our  Arabic   system  of 


MULTIPLICATION.  225 

notation  and  the  decimal  method  of  numeration,  it  is  not  neces- 
sary that  the  elementary  products  should  extend  beyond  "  nine 
times."  It  is  not  at  all  inconvenient,  however,  but  quite  nat- 
ural that  it  should  include  eleven  and  twelve  times,  since  the 
names  eleven  and  twelve  are  a  seeming  departure  from  the  dec- 
imal system  of  numeration. 

Case  II.  To  multiply  riumbers  expressed  by  the  Arabic 
system  of  notation.  When  the  numbers  are  small,  as  we  have 
seen,  we  multiply  them  directly  as  wholes;  when  we  extend 
beyond  the  elementary  products,  the  principle  is  to  multiply 
by  parts.  Thus,  instead  of  multiplying  the  multiplicand  as  a 
single  number,  we  multiply  first  one  group,  then  the  next  group, 
and  so  on,  as  we  united  numbers  in  addition.  Also,  when  the 
multiplier  exceeds  nine — or  in  practice,  twelve — that  is,  when 
it  is  expressed  in  two  or  more  places,  we  multiply  first  by  the 
units  term,  then  by  the  tens  term,  etc.;  and  then  take  the  sum 
of  these  partial  products. 

To  illustrate,  let  it  be  required  to  multiply  65  by  37.  To 
multiply  by  thirty-seven  as  a  single  number,  would  be  quite  a 
difficult  task.  We  do  not  attempt  this,  however,  but  first  mul- 
tiply by  7  units,  one  part  of  37,  and  then  by  3  tens,  the  other 
part  of  37,  and  then  take  the  sum  of  these  products.  It  is  also 
seen  that  the  number  65  is  not  multiplied  as  a  single  number, 
but  by  using  its  parts,  5  units  and  6  tens.  The  method  of 
explaining  the  process  is  as  follows: 

Solution. — Thirty-seven  times   65   equals   7     operation. 
times  65  plus  3  tens  times  65.     Seven  times  5  65 

units  are  35  units,  or  3  tens  and  5  units ;  we  ^ 

write  the  5  units,  and  reserve  the  3  tens  to  add  455 

to  the  product  of  tens.     Seven  times  6  tens 
are  42  tens,  which,  increased  by  3  tens,  equals  2±Q& 

45   tens,  or  5  tens  and   4   hundreds,  which  we   write  in  its 
proper  place.     Multiplying  similarly  by  3  tens,  we  have  5  tens 
9  hundreds  and  1  thousand;  and  taking  the  sum  of  these  two 
partial  products,  we  have  2405. 
15 


226  THE    PHILOSOPHY    OF    ARITHMETIC. 

This  method  of  multiplication  is  founded  upon,  and  is  only 
possible  with  a  system  of  notation  similar  to  the  Arabic. 
Without  some  such  method  of  expressing  numbers  in  char- 
acters, the  multiplication  of  large  numbers  would  be  exceed- 
ingly laborious,  if  not  altogether  impossible.  We  are  thus 
continually  reminded  of  the  advantages  of  the  Arabic  system 
of  notation,  and  learn  almost  to  venerate  the  people  and 
country  that  conferred  so  great  a  boon  upon  the  human  race 
by  its  invention. 

Rule. — The  error  of  confounding  the  meaning  of  figure  and 
number  is  repeated  in  the  rule  for  multiplication.  The  rule,  as 
usually  given  is,  "Multiply  each  figure  of  the  multiplicand  by 
the  multiplier,"  etc., or  "Multiply  the  multiplicand  by  each  figure 
of  the  multiplier,"  etc.  This  error  is  easily  avoided  by  the  use  of 
the  word  term  for  figure.  It  should  be  remembered  that  we  have 
two  distinct  things,  the  number  and  the  numerical  expression. 
The  parts  of  the  numerical  expression  are  figures ;  the  parts  of  the 
entire  number  are  numbers.  The  word  term  may  be  employed 
to  express  both  of  these,  without  any  obscurity  and  with  much 
convenience.  The  rule  will  then  read,  "Multiply  each  term  of 
the  multiplicand  by  the  multiplier,"  etc.,  or,  "by  each  term  of 
the  multiplier,"  etc. 

Remark. — We  write  the  numbers  as  indicated  above  for  con- 
venience in  multiplying.  The  placing  of  the  multiplier  under 
the  multiplicand,  instead  of  over  it,  and  multiplying  from 
below,  is  a  mere  matter  of  custom,  corresponding  with  the 
method  of  adding  and  subtracting.  We  begin  at  the  right 
hand  to  multiply  so  that  when  any  product  exceeds  nine,  we 
may  incorporate  the  number  expressed  by  the  left  hand  figure 
with  the  following  product.  The  convenience  of  this  will  be 
readily  appreciated  by  performing  the  multiplication  by  begin- 
ning at  the  left.  It  was  formerly  the  custom,  however,  to 
begin  at  the  left,  writing  the  partial  products  in  their  order  and 
subsequently  collecting  them. 


CHAPTER  IV 

DIVISION. 

rpiIE  general  process  of  analysis  is  Subtraction.  After  the 
J-  mind  becomes  familiar  with  this  general  process,  it  begins 
to  extend  and  specialize  it,  and  thus  arises  a  new  process  called 
Division.  Division  is,  therefore,  a  special  case  of  subtraction, 
in  which  the  same  number  is  to  be  successively  subtracted  with 
the  object  of  finding  how  many  times  it  is  contained.  The  idea 
of  Division  is  thus  seen  to  be  contained  in  that  of  Subtraction, 
and  is  the  outgrowth  of  it. 

Division  may  also  be  regarded  as  arising  from  a  reversing 
of  the  process  of  multiplication.  In  multiplication,  we  obtain 
the  product  of  two  numbers  ;  and  since  the  product  is  a  number 
of  times  the  multiplicand,  we  may  regard  it  as  containing 
the  multiplicand  a  number  of  times.  Thus,  since  four  times 
five  are  twenty,  twenty  may  be  considered  as  containing 
five,  four  times.  Division  is  thus  regarded  as  an  analytic 
process,  arising  from  reversing  the  synthetic  process  of  multi- 
plication. 

It  thus  appears  that  Division  may  have  originated  in  either 
of  two  different  ways.  In  which  way  it  did  actually  arise,  it 
is  impossible  for  us  to  decide  with  certainty.  It  has  generally 
been  supposed,  judging  from  the  old  definition  that  "  Division 
is  a  concise  method  of  Subtraction,"  that  it  had  its  genesis  in 
Subtraction.  My  own  opinion,  however,  is  that  it  originated 
by  reversing  multiplication,  for  which  I  state  the  following 
reasons: — First,  as  subtraction  arose  from  reversing  the  pro- 

(227) 


228  THE   PHILOSOPHY    OF   ARITHMETIC. 

cess  of  addition,  so  is  it  natural  to  suppose  that  division,  a 
concise  subtraction,  would  arise  from  reversing  multiplica- 
tion, a  concise  addition.  Second,  division  involves,  as  essen- 
tial to  it,  the  idea  of  "times,"  which  had  already  appeared 
in  multiplication.  It  seems  much  more  natural  to  take  the 
idea  of  times  from  multiplication,  where  it  already  existed,  than 
to  originate  it  from  the  process  of  subtraction. 

Definition. — Division  is  the  process  of  finding  the  quotient 
of  two  numbers.  The  quotient  of  two  numbers  is  the  number 
of  times  that  one  number  contains  the  other.  The  number 
divided  is  the  Dividend ;  the  number  we  divide  by  is  the 
Divisor.  The  definition  usually  given  is,  "Division  is  the 
process  of  finding  how  many  times  one  number  is  contained  in 
another."  This  is  regarded  as  correct,  but  is  less  simple  and 
concise  than  the  one  above  suggested. 

Defining  division  in  this  manner,  we  have  a  simple  and  con- 
cise definition,  easily  understood  and  logically  accurate.  It 
follows  the  method  generally  adopted  for  addition  and  sub- 
traction, and  which  I  have  also  suggested  for  multiplication^ 
and  presents  a  happy  uniformity  in  the  definitions  of  the  four 
fundamental  operations  of  arithmetic.  The  objects  of  these 
four  fundamental  processes,  as  thus  presented,  will  respectively 
be  to  find  the  Sum,  the  Difference,  the  Product,  and  the  Quo- 
tient of  numbers. 

Principles. — The  operations  in  division  are  controlled  by 
certain  necessary  laws  of  thought  to  which  we  give  the  name 
of  principles.  The  following  are  the  most  important  of  the 
principles  of  division: 

1,  The  dividend  and  divisor  are  always  similar  numbers. 
This  is  true  of  division  scientifically  considered,  as  may  be 
seen  by  regarding  it  as  originating  in  subtraction  or  multipli- 
cation. Supposing  that  it  has  its  root  in  subtraction,  and 
remembering  that  in  subtraction  the  two  terms  must  be  alike, 
we  see  that  this  principle  follows  of  necessity.  Thus,  if  w,e 
inquire  how  many  times  one  number  is  contained  in  another,. 


division.  229 

it  is  evident  that  these  numbers  must  be  similar.  We  may  inquire 
how  many  times  4  apples  are  contained  in  8  apples,  but  not 
how  many  times  4  peaches  are  contained  in  8  apples.  Neither 
can  we  say  "How  many  times  is  4  contained  in  8  apples?"  for  8. 
apples  will  not  contain  the  abstract  number  4  any  number  of 
times.  The  same  conclusion  is  reached  if  we  regard  division 
as  originating  in  multiplication.  If  we  assume  that  4  is  con- 
tained in  8  apples  2  apples  times,  it  would  follow  that  2  apples 
times  4  equals  8  apples,  which  is  absurd. 

Several  recent  writers  take  the  position  that  a  concrete  number 
may  be  divided  by  an  abstract  number,  because  in  practice  we 
hus  divide  a  concrete  number  into  equal  parts.  This  is  a 
subordination  of  science  to  practice,  which  is  neither  philo- 
sophical nor  necessary.  The  practical  case  which  they  thus 
try  to  include  in  the  theory  of  the  subject,  admits  of  a  scientific 
and  simple  explanation,  without  any  modification  of  the  funda- 
mental idea  of  division  ;  and  when  thus  explained  it  becomes 
apparent  that  the  two  terms  are  similar  numbers. 

2.  The  quotient  is  always  an  abstract  number.  This  results 
from  the  fundamental  idea  of  division,  whether  we  regard  it  as 
originating  in  subtraction  or  multiplication.  The  quotient  shows 
how  many  times  one  number  is  contained  in  another,  and  one 
number  cannot  be  contained  in  another  number  yards  times,  or 
apples  times,  etc.,  from  which  it  follows  that  the  quotient 
must  be  abstract.  The  quotient  shows  how  many  times  one 
number  may  be  subtracted  from  or  taken  out  of  another  before 
exhausting  the  latter,  and  must  therefore  be  a  number  of  times, 
and  consequently  abstract.  Or,  regarding  it  as  arising  from 
multiplication,  the  quotient  is  the  number  of  times  the  divisor 
which  equals  the  dividend ;  and,  as  such,  is  a  multiplier ;  and 
must,  consequently,  be  abstract.  Suppose  it  were  said  that  2 
is  contained  in  8  apples,  "4  apples  times," — and  all  authors 
agree  as  to  the  quotient  denoting  the  number  of  times  the 
divisor  is  contained  in  the  dividend — then  it  would  follow  that 
"4  apples  times"  2   are  8  apples ;  which  is,  of  course,  absurd. 


230  THE    PHILOSOPHY    OF    ARITHMETIC. 

3.  The  remainder  is  always  similar  to  the  dividend.  This 
is  evident,  since  the  remainder  is  an  undivided  part  of  the  divi- 
dend. In  practice,  as  above  intimated,  some  of  these  princi- 
ples seem  to  be  violated,  but  if  the  analysis  be  given,  it  will  be 
seen  that  the  violation  is  merely  seeming,  and  not  actual. 

4.  The  following  principles  show  the  relation  of  the  terms 
in  division: 

1.  The  dividend  equals  the  product  of  the  divisor  and  quo- 
tient. 

2.  The  divisor  equals  the  quotient  of  the  dividend  and 
quotient. 

3.  The  dividend  equals  the  product  of  the  divisor  and  quo- 
tient, plus  the  remainder. 

4.  The  divisor  equals  the  dividend  minus  the  remainder, 
divided  by  the  quotient. 

5.  The  following  principles  show  the  result  of  multiplying 
or  dividing  the  terms  in  division: 

1.  Multiplying  the  dividend  or  dividing  the  divisor  by  any 
number  multiplies  the  quotient  by  that  number. 

2.  Dividing  the  dividend  or  multiplying  the  divisor  by  any 
number  divides  the  quotient  by  that  number. 

3.  Multiplying  or  dividing  both  divisor  and  dividend  by  the 
same  number  does  not  change  the  quotient. 

Cases. — Division  is  philosophically  divided  into  two  general 
cases.  The  first  case  consists  in  finding  the  quotient  of  num- 
bers independently  of  the  method  of  notation  used  to  express 
them.  The  second  case  is  that  which  grows  out  of  the  use  of 
the  Arabic  system  of  notation.  The  former  case  deals  with 
small  numbers  mentally,  and  may  be  called  Mental  Division; 
the  latter  deals  with  large  numbers,  expressed  by  means  of 
written  characters,  and  may  be  called  Written  Division. 
The  former  is  an  independent  process,  belonging  to  pure  num- 
ber, and  is  complete  in  itself;  the  latter  operates  by  means  of 
the  Arabic  characters,  and  is  dependent  upon  the  former  for  its 
elementary  quotients. 


DIVISION.  231 

Method. — In  division  we  first  find  the  elementary  quotients 
corresponding  to  the  elementary  products  of  the  multiplication 
table.  These  may  be  obtained  in  two  different  ways,  as  will 
be  explained.  In  the  second  case  we  operate  by  parts,  using 
the  elementary  quotients  as  a  basis  of  operation.  The  two 
cases  will  be  formally  presented. 

Case  I.  To  find  the  elementary  quotients  of  arithmetic. 
The  first  object  in  division  is  to  find  the  elementary  quotients 
corresponding  to  the  elementary  products  of  the  multiplication 
table.  These  quotients  admit  of  a  double  origin  ;  that  is,  they 
may  be  derived  by  the  method  of  concise  subtraction,  or  of 
reverse  multiplication.  Thus,  if  we  wish  to  ascertain  how 
many  times  five  is  contained  in  twenty,  we  may  find  how  many 
times  five  can  be  taken  out  of  twenty  by  subtraction,  and  this 
will  show  how  many  times  twenty  contains  five.  This  is  the 
method  of  subtraction,  and  as  thus  viewed,  division  may  be 
regarded  as  a  method  of  concise  subtraction.  Again,  since  we 
know  that  four  times  five  are  twenty,  we  can  immediately 
infer  that  twenty  contains  four  fives,  or  that  twenty  contains 
five  four  times.  This  is  the  method  of  multiplication,  and  as 
thus  viewed,  division  may  be  regarded  as  a  method  of  reverse 
multiplication. 

Either  of  these  two  methods  may  be  used  for  finding  the 
elementary  quotients,  but  the  method  of  reverse  multiplication 
is  much  more  convenient  in  practice.  The  quotients  are  imme- 
diately derived  from  the  products  of  the  multiplication  table, 
and  we  are  thus  saved  the  labor  of  forming  and  committing  a 
table  of  division.  If,  however,  the  elementary  quotients  be 
derived  by  subtraction,  it  will  be  necessary  to  construct  a 
division  table,  and  commit  the  quotients,  as  we  do  the  products 
in  multiplication. 

These  elementary  quotients,  whether  derived  by  multiplica- 
tion or  subtraction,  are  the  result  of  a  process  of  reasoning. 
The  process  of  thought  may  be  illustrated  in  the  problem,  "Five 
is  contained  how  many  times  in  twenty?"  and  is  as  follows: 


232  THE    PHILOSOPHY    OF    ARITHMETIC. 

Five  is  contained  as  many  times  in  twenty  as  twenty  is  times 
five;  but  twenty  is  four  times  five;  hence,  Jive  is  contained  in 
twenty,  four  times.  In  ordinary  language,  this  is  abbreviated 
thus:  five  is  contained  four  times  in  twenty,  since  four  times 
five  are  twenty. 

By  the  method  of  subtraction  we  reason  thus:  five  is  con- 
tained as  many  times  in  twenty  as  five  can  be  successively  sub- 
tracted from  or  taken  out  of  twenty ;  but  five  can  be  suc- 
cessively subtracted  from  twenty,  four  times;  hence,  five  is 
contained/bur  times  in  twenty.  The  ordinary  form  of  thought 
is,  five  is  contained  four  times  in  twenty,  since  it  can  be  sub- 
tracted from  twenty,  four  times.  By  "subtracted  from,"  as  here 
used,  we  mean  subtracted  successively  from  until  twenty  is 
exhausted. 

Case  II.  To  divide  when  the  numbers  are  expressed  in 
the  Arabic  scale  of  notation.  When  the  numbers  are  small, 
we  divide  them,  as  we  have  seen,  directly  as  wholes;  when  we 
extend  beyond  the  elementary  quotients,  the  principle  is  to 
divide  by  parts.  The  dividend  is  not  immediately  divided  as 
a  whole,  but  is  regarded  as  consisting  of  parts  or  groups;  and 
these  are  so  divided  that,  when  remainders  occur,  they  may  be 
incorporated  with  inferior  groups,  and  thus  the  whole  number 
be  divided.  This  method,  as  in  multiplication,  is  due  to  the 
system  of  Arabic  notation,  and  enables  us  to  divide  large  num- 
bers, which  would  be  exceedingly  difficult,  if  not  impossible, 
with  a  different  system  of  notation. 

In  Written  Division,  or  division  of  large  numbers,  two 
cases  are  presented.  First,  when  the  divisor  is  so  small  that 
only  the  elementary  dividends  and  divisors  are  used ;  second, 
when  the  divisors  and  dividends  are  larger  than  those  employed 
in  obtaining  the  elementary  quotients.  The  methods  of  treat- 
ing these  two  cases  are  distinguished  as  Short  Division  and 
Long  Division.  In  Short  Division,  the  partial  dividends  are 
not  written;  in  Long  Division,  the  partial  dividends  and  other 
necessary  work  nre  written. 


DIVISION. 


233 


Illustration.— To  illustrate  the  method  of  Short  Division, 
divide  537  by  3.  Here  we  cannot  divide  the  given  number  as 
a  whole,  that  is,  as  five  hundred  and  thirty-seven,  but  by  sep- 
arating it  into  parts,  we  can  readily  divide  these  parts,  as  they 
give  only  the  elementary  quotients.  Thus,  we  first  divide 
five  hundred,  reduce  the  remainder  of  the  group  to  tens  and 
incorporate  with  the  tens  group,  making  23  tens,  divide  this  as 
before,  and  thus  continue  until  the  whole  of  the  number  hate 
been  divided. 

When  the  divisor  is  greater  than  12,  the  division  can  no 
longer  be  performed  by  using  the  elementary  dividends  and 
quotients.  The  process  then  becomes  more  difficult,  although 
it  involves  the  same  principles  as  when  smaller  numbers  are 
used.  As  the  elementary  quotients  were  derived  from  multipli- 
cation, so  in  Long  Division  we  determine  the  quotient  by  mul- 
tiplying. We  multiply  the  divisor  by  some  number  which 
we  suppose  to  be  the  quotient  term,  and  if  the  product  does 
not  exceed  the  partial  dividend,  nor  the  difference  between  the 
product  and  partial  dividend  exceed  the  divisor,  we  know  that 
we  have  obtained  the  correct  quotient  figure.  The  method 
described  is  so  common  that  it  need  not  be  illustrated  by  a 
problem. 

Bule.— -The  mistake  of  using  figure  for  number  is  also  made 
in  stating  the  rule  for  division.  One  author  says,  "  Find  how 
many  times  the  divisor  is  contained  in  the  fewest  figures  on 
the  left  of  the  dividend,"  etc.;  another  says,  "Take  for  the  first 
partial  dividend  the  fewest  figures  of  the  given  dividend," 
etc.;  another  says,  "Take  for  the  first  partial  dividend  the 
least  number  of  figures  on  the  left  that  will  contain  the  divisor," 
etc.  Of  course,  figures  will  not  contain  the  divisor;  the  num- 
ber expressed  by  the  figures  is  what  is  intended,  and  therefore 
should  be  expressed.  The  error  may  be  corrected  by  saying, 
"Divide  the  number  expressed  by  the  fewest  figures  on  the 
left  that  will  contain  the  divisor,"  or,  "by  the  fewest  terms,'' 
etc 


234  THE    PHILOSOPHY    OF    ARITHMETIC. 

Remark. — We  write  the  divisor  at  the  left  of  the  dividend 
and  the  quotient  at  the  right  as  a  matter  of  custom.  Some  pre- 
fer writing  the  divisor  at  the  right  and  placing  the  quotient 
under  the  divisor.  We  begin  at  the  left  to  divide,  so  that  the 
remainder,  when  one  occurs,  may  be  united  with  the  number 
of  units  of  the  next  lower  order,  giving  a  new  partial  divi- 
dend. If  we  attempt  to  divide  by  beginning  at  the  right,  we 
will  see  the  advantage  of  the  ordinary  method. 


SECTION    II. 

DERIVATIVE  OPERATIONS  OF  SYNTHESIS 
AND  ANALYSIS. 


I.  Introduction. 


II.  Composition. 


III.  Factoring. 


IV.  Common  Divisob. 


V.  Common  Multiple 


VI.  Involution. 


VII.  Evolution. 


CHAPTER  I. 

INTRODUCTION    TO    DERIVATIVE    OPERATIONS. 

THE  four  Fundamental  Operations  are  the  direct  and  imme- 
diate outgrowth  of  the  general  processes  of  synthesis  and 
analysis  as  applied  to  numbers.  They  are  called  Fundamental 
Operations  because  all  the  other  operations  involve  one  or 
more  of  these,  and  may  be  regarded  as  being  based  upon  them. 
They  are  the  foundation  or  basis  upon  which  the  others  are 
built  up,  the  germ  from  which  they  are  evolved,  the  soil  out  of 
which  they  grow. 

Several  of  the  processes  of  arithmetic  are  so  intimately 
related  to  the  fundamental  operations  that  they  may  be 
regarded  as  directly  originating  in  and  growing  out  of  them. 
Such  are  the  processes  of  Factoring,  Common  Multiple,  Com- 
mon Divisor,  etc.  These  processes  have  their  roots  in  the 
general  notions  cf  the  fundamental  operations,  and  are 
evolved  from  them  by  a  modification  and  extension  of  the  pri- 
mary analytic  and  synthetic  processes.  They  are  developed 
by  the  thought  process  of  comparison,  though  they  have  not 
their  basis  in  comparison,  like  the  processes  of  Ratio,  Propor- 
tion, etc.  Being  thus  derived  from  the  fundamental  operations, 
they  may  be  called  the  Derivative  Operations  of  synthesis  and 
analysis.  Let  us  notice  the  origin  and  nature  of  these  deriva- 
tive operations. 

If  two  or  more  numbers  are  multiplied  together,  and  the 
result  is  considered  with  respect  to  its  elements,  we  have  the 
idea  of  a  Composite  Number.  The  general  process  of  forming 
composite  numbers  may  be  called  Composition.     The  numbers 

(237) 


238  THE    PHILOSOPHY    OF    ARITHMETIC. 

synthetized  in  forming  a  composite  number  are  called  Factors 
of  that  number.  If  we  form  a  composite  number  consisting 
of  two  equal  factors,  we  have  a  square ;  of  three  equal  factors, 
a  cube,  etc.,  and  the  process  is  called  Involution.  If  we  find 
a  composite  number  which  is  a  number  of  times  each  of  several 
numbers,  or  is  so  composed  that  each  of  them  is  one  of  its 
factors,  it  is  called  a  common  multiple  of  these  numbers,  and 
the  process  is  known  as  finding  Common  Multiples. 

These  processes  are  distinct  from  Multiplication,  though 
related  to  it.  They  employ  multiplication  and  are  the  out- 
growth of  the  general  multiplicative  idea,  but  pass  beyond  the 
primary  idea  of  multiplication.  In  multiplication,  the  main 
idea  is  the  operation  of  repeating  one  number  as  many  times 
as  there  are  units  in  another  to  obtain  a  result ;  here  the  thought 
is  the  result  of  the  operation  compared  with  the  numbers 
multiplied  together.  In  the  former  case,  the  process  is  purely 
synthetic  ;  here  comparison  unites  with  synthesis,  and  employs 
it  for  a  particular  object.  The  operation  of  multiplying  is 
assumed  as  a  fact,  and  employed  for  the  purpose  of  attaining 
a  result  bearing  some  relation  to  the  elements  combined. 

Having  obtained  composite  numbers,  and  the  idea  of  their 
being  composed  of  factors,  we  naturally  begin  to  analyze  them 
into  their  elements  in  order  to  discover  these  factors.  This 
gives  rise  to  an  analytic  process,  the  converse  of  Composition. 
The  general  process  of  analyzing  a  number  into  its  factors  is 
called  Factoring.  If  we  resolve  a  number  into  several  equal 
factors  for  the  purpose  of  seeing  what  factor  must  be  repeated 
two,  or  three,  etc.,  times  to  produce  the  number,  we  have  a 
process  known  as  Evolution.  If  we  have  given  several  num 
bers,  and  proceed  to  find  a  common  factor  of  these  numbers, 
we  have  the  process  known  as  Common  Divisor. 

These  processes,  though  related  to  Division,  are  clearly  dis- 
tinguished from  it.  They  are  an  outgrowth  of  the  general 
idea  of  division,  but  extend  beyond  it.  In  division  it  is  the 
operation  of  finding  how  manv  times  one  number  is  contained 


INTRODUCTION. 


239 


in  another  that  is  the  prominent  idea;  here  the  idea  is  the 
result  considered  in  relation  to  the  number  or  numbers 
operated  upon.  In  Factoring,  the  process  of  comparison 
enters  as  an  important  element.  Division  is  a  process  purely 
analytical ;  Factoring  is  analysis,  and  more ;  it  is  analysis  plus 
comparison.  It  has  its  root  in  Analysis,  and  is  developed  by 
the  thought-process  of  Comparison. 

There  are,  therefore,  two  general  derivative  processes,  Com- 
position and  Factoring,  each  of  which  embraces  corresponding 
and  opposite  processes.  The  terms,  Composition  and  Factor- 
ing, are  in  practice  restricted  to  the  general  processes ;  the 
special  processes  are  known  by  their  particular  names.  We 
have  thus  three  pairs  of  derivative  processes, —  Composition 
and  Factoring,  Multiples  and  Divisors,  and  Involution  and 
Evolution.     These  will  be  treated  in  successive  chapters. 


CHAPTER  II 

COMPOSITION. 

COMPOSITION  is  the  process  of  forming  composite  num- 
bers when  their  factors  are  given.  It  is  a  general  process 
which  contains  several  subordinate  and  special  ones.  When 
fully  analyzed,  it  will  be  seen  to  present  several  interesting 
cases  besides  the  more  particular  ones  of  Involution  and  Mul- 
tiples. From  the  previous  analysis  it  is  seen  that  there  is  a 
real  case  of  Synthesis,  the  converse  of  the  analytic  process 
of  Factoring. 

This  new  generalization,  and  the  term  I  have  applied  to  it, 
will,  I  trust,  receive  the  approval  of  mathematicians.  Its 
importance  as  a  logical  necessity,  is  seen  in  its  relation  to 
Factoring.  In  the  fundamental  operations  each  synthetic  pro- 
cess has  its  corresponding  analytic  process.  Thus,  addition  is 
synthetic,  subtraction  is  analytic ;  multiplication  is  synthetic, 
division  is  analytic.  It  follows,  therefore,  that  there  should 
be  a  synthetic  process  corresponding  to  the  analytic  process 
of  Factoring.  This  process  I  have  presented  under  the  name 
of  Compositio?i,  or  the  process  of  forming  composite  numbers. 

Cases. — There  are  several  interesting  and  practical  cases  of 
Composition,  some  of  the  most  important  of  which  are  the 
following: 

I.  To  form  a  composite  number  out  of  any  factors. 

II.  To  form  a  composite  number  out  of  equal  factors. 

III.  To  form  a  composite  number  out  of  factors  bearing  any 
definite  relation  to  each  other. 

(240) 


COMPOSITION.  24-1 

IV.  To  form  composite  numbers  which  have  one  or  more 
given  common  factors. 

V.  To  form  several  or  all  of  the  composite  numbers  possible 
out  of  given  factors. 

VI.  To  determine  the  number  of  composite  numbers  that 
can  be  formed  out  of  given  factors. 

Method  of  Treatment. — The  method  of  treatment  is  to  com- 
bine these  factors  by  multiplication  in  such  a  manner  as  to 
attain  the  result  desired.  I  will  briefly  state  the  manner  of 
treating  each  case. 

Case  I.  To  form  a  composite  number  out  of  any  factors 
In  Case  I.  we  find  the  result  by  simply  taking  the  product  of 
the  factors.  Thus  the  composite  number  formed  from  the  fac- 
tors 2,  3,  and  4  equals  2x3x4,  or  24. 

Case  II.  To  form  a  composite  number  out  of  equal  fac- 
tors. Case  II.  may  be  solved  in  the  same  manner  as  Case  I.,  or 
we  may  multiply  a  partial  result  by  itself  or  by  another  partial 
result,  to  obtain  the  entire  result.  Thus,  if  we  wish  to  find 
the  composite  number  consisting  of  eight  2's,  we  may  multi- 
ply 2  by  2,  giving  4,  then  multiply  4  by  4,  giving  16,  and  then 
multiply  16  by  16,  giving  256,  the  number  required. 

Case  III.  To  form  a  composite  number  out  of  factors 
bearing  any  definite  relation  to  each  other.  In  this  case  we 
may  have  given  one  factor  and  the  relation  of  the  other  factors 
to  it ;  we  first  find  the  factors  and  then  take  their  product. 
Thus,  required  the  number  consisting  of  three  factors,  the  first 
being  4,  the  second  twice  the  first,  and  the  third  three  times  the 
second.  Here,  we  first  find  the  second  factor  to  be  8,  and  the 
third  to  be  24,  and  then  take  the  product  of  4,  8,  and  24,  which 
we  find  to  be  768. 

Case  IV.  To  form  composite  numbers  which  have  one  or 
more  given  common  factors.  This  case  may  be  solved  by  tak- 
ing the  given  common  factor,  and  multiplying  it  by  any  other 
factors  we  choose.  If  it  is  required  that  the  factor  given  be 
the  largest  common  factor  of  the  numbers  obtained,  the  mul- 
tipliers selected  must  be  prime  to  each  other.  To  illustrate, 
16 


242  THE    PHILOSOPHY   OF    ARITHMETIC. 

find  three  numbers  whose  largest  common  factor  shall  be  12. 
If  we  multiply  12  by  2,  4,  and  6,  we  will  have  24,  48,  and  72, 
three  numbers  whose  common  factor  is  12;  but  since  the  num- 
bers used  as  multipliers  have  a  common  factor,  12  is  not  the 
largest  factor  common  to  these  three  numbers.  To  find  three 
numbers  having  12  as  their  largest  common  factor,  we  may 
multiply  12  by  2,  3,  and  5,  which  gives  us  the  numbers  24,  36, 
and  60,  in  which  12  is  the  largest  common  factor. 

Case  V.  To  form  several  or  all  of  the  composite  numbers 
possible  out  of  given  factors.  In  this  case  we  may  take  the 
factors  two  together,  three  together,  etc.,  until  they  are  taken 
all  together;  or  we  may  multiply  1  and  the  first  factor  by  1 
and  the  second  factor,  the  products  thus  obtained  by  1  and  the 
third  factor,  etc.,  until  all  the  factors  are  used.  To  illustrate, 
form  all  the  possible  composite  numbers  out  of  2,  3,  5,  and  7. 

We  first  find  all  the  possible  pro- 
ducts   taking    them    two    together;  operation. 

then  all    the    products   taking    them     «*?=?A     o^=i'? 

JXO=10      oX  7  =  ^1 
three  together,  and  then  the  products     2x7  =  14     5x  7=35 

taking    them    four    together,    as    is  2x3x5=30 

shown    in     the    margin.       Another  2x3x7  =  42 

method,    not     quite     so    simple    in  q     r     7— in* 

thought    but     more    convenient    in       2x3x5x7=210 
practice,  is  as  follows: 

Multiplying  1  and  2  by  1  and  3,  will  give  1,  2,  3,  and  all  the 
composite  numbers  that  can  be  formed  out  of  2  and  3;  these 
multiplied 

by  1  and  5     ^  operation. 

will  give  1, 
2,  3,  5,  and 
allthecom- 


1  2 
1  3 


12  3  6 

1  5 

positenum-     x  2  3  5  6  10  15^0 
bers   that     |  7 

can       be     1  2  3  5  6  10  15  30  7  14  21  42  35  70  105  210 
formed  out 

of  2,  3,  and  5 ;  these  multiplied  by  1  and  7  will  give  1,  2,  3,  5, 


COMPOSITION.  243 

7,  and  all  the  composite  numbers  that  can  be  formed  out  of  2, 

3,  5,  and  7.     Omitting  1,  2,  3,  5,  and  7  in  the  last  result,  and 

we  have  all  the  composite  numbers  that  can  be  formed  out  of 

2,  3,  5,  and  7. 

If  some  of  the  given  factors  are  alike,  we  have  an  interesting 

modification  of  this  case.     Thus,  suppose  we  wish  to  find  the 

composite    numbers    which 

can  be  composed  out  of  2,  2,  operation. 

2,  3,  and  3.    In  this  problem     J  |  4  8 

since  2  is  used  three  times    

,     .,     n    .       .         1  2  3  4  6  8  9  12  18  24  36  72 
we  may  make  the  first  series 

1,  2,  22,  and  23,  or  1,  2,  4,  and  8;  and  since  3  is  used  twice,  the 

second  series  will  be  1,  3,  and  32,  or  1,  3,  and  9 ;    and  the 

products  of  these,  omitting  1,  2,  and  3,  will  be  the  composite 

numbers  required. 

Case  VI.  To  determine  the  number  of  composite  num- 
bers that  can  be  formed  out  of  given  factors.  We  may  solve 
this  case  by  increasing  the  number  of  times  each  factor  is  used 
by  unity,  take  the  product  of  the  results  and  diminish  it  by 
the  number  of  different  factors  used  increased  by  one.  The 
reason  for  this  method  may  be  readily  shown.  Suppose  we 
wish  to  find  how  many  composite  numbers  can  be  formed  with 
three  2's  and  two  3's. 

Here  we  see  that  2  used  three  times  as  a  factor  gives  with  1 
a  series  of  four  terms ;  and  3  used  twice  as  a  factor  gives 
with  1  a  series  of  three  terms ;  hence  the  product  will  give  a 
series  of  4x3  or  12  terms,  and  omitting  the  unit  and  2  and  3, 
we  have  nine  terms.  The  inference  from  this  so'ution  will 
give  the  method  stated  above. 


CHAPTER  IIL 

FACTORING. 

FACTORING  is  the  process  of  finding  the  factors  of  com- 
posite  numbers.  It  is  the  reverse  of  Composition.  In 
Composition  we  have  given  the  factors  to  find  the  number ;  in 
Factoring  we  have  given  the  number  to  find  the  factors.  Com- 
position is  a  synthetic  process ;  it  proceeds  from  the  parts  by 
multiplication  to  the  whole.  Factoring  is  an  analytic  process ; 
it  proceeds  from  the  whole  by  division  to  the  parts. 

A  Factor,  as  now  generally  presented  in  arithmetic,  is 
regarded  as  a  divisor  of  a  number,  rather  than  a  maker  or  pro- 
ducer of  the  number.  This  I  regard  as  an  error.  The  origin 
of  the  word,  facio,  I  make,  indicates  its  original  meaning  to  be 
a  maker  of  a  composite  number.  The  fact  of  a  Factor 
of  a  number  being  a  divisor  of  it  is  a  derivative  idea,  re- 
sulting from  the  primary  conception  of  its  entering  into  the 
composition  of  the  number.  This  primary  idea  of  the  office  of 
a  Factor  is  the  one  that  should  be  primarily  presented  to  pupils, 
rather  than  the  secondary  or  derivative  idea.  We  should 
define  according  to  the  fundamental,  rather  than  the  derivative 
office.  To  do  otherwise  is  to  invert  the  logical  relation  of  ideas, 
and  must,  as  I  have  known  it,  tend  to  confusion.  Thus  taught, 
it  is  seen  that  the  proposition,  a  factor  of  a  number  is  a  divi- 
sor of  the  number,  is  an  immediate  inference,  which  would  have 
to  be  inverted  if  the  secondary  office  of  a  factor  is  made  the 
fundamental  idea. 

(244) 


FACTORING. 


245 


Cases.— Factoring  presents  several  cases  analogous  to  those 
of  Composition.  Some  of  the  principal  ones  are  the  following, 
which,  it  will  be  noticed,  are  the  correlatives  of  those  givei 
under  Composition. 

I.  To  resolve  a  number  into  its  prime  factors. 

II.  To  resolve  a  number  into  equal  factors, 

III.  To  resolve  a  number  into  factors  bearing  a  certain  rela- 
tion to  each  other. 

IV.  To  find  the  divisors  common    to  two  or  more   num- 
bers. 

Y.  To  find  all  the  factors  or  divisors  of  a  number. 
YI.  To  find  the  number  of  divisors  of  a  number. 
Method.— The  general  method  of  treatment  is  to  resolve  the 
number  or  numbers  into  their  prime  factors,  and  then  combine 
these  factors  when  necessary  so  as  to  give  the  required  result. 
The  prime  factors  of  a  number  are  found  by  division,  and  con- 
sequently it  is  convenient  to  know  before  trial  what  numbers 
are  composite  and  can  be  factored,  and  the  conditions  of  their 
divisibility.     Hence,  the  subject  of  Factoring  gives  rise  to  the 
investigation  of  the  methods  of  determining  prime  and  com- 
posite numbers,  and  the  conditions  of  the  divisibility  of  com- 
posite numbers.     This  subject  will  be  treated  under  the  head 
of  Prime  and  Composite  Numbers.     The  method  of  treating 
each  of  the  above  named  cases  of  factoring  will  be  briefly  stated. 
Case  I.   To  resolve  a  number  into  its  prime  factors.     In 
Case  I  we  divide  the  number  by  any  prime  number  greater 
than    1   which   will   exactly  divide   it;   divide   the  quotient, 
if  composite,  in  the  same  manner;  and  thus  continue  until  the 
quotient  is  prime.     The  divisors  and  the  last  quotient  will  be 
the  prime  factors  required. 

Thus,  suppose  we  have  given  105  to  find  its  prime 
factors.     Dividing  105  by  the   prime  factor  3,  and     3^105 
the  quotient  35  by  5,  we  see  that  105  is  composed      5;j55 
of  the  three  factors  3,  5,  and  t,  and   since  these  are 
prime  numbers,  its  prime  factors  are  3,  5,  and  1. 


246  THE    PHILOSOPHY    OF    ARITHMETIC. 

Case  II.  To  resolve  a  number  into  equal  factors.  In  Case 
II.  we  resolve  the  number  into  its  prime  factors  and  then  com- 
bine by  multiplication  one  from  each  set  of  two  equal  factors, 
when  we  wish  one  of  the  two  equal  factors  of  the  number ;  one 
from  each  set  of  three  equal  factors  when  we  wish  one  of 
three  equal  factors,  etc. 

Thus,  suppose  we  wish  to  find  the  (2x2x2x 

three  equal  factors  of  216,  or  one  of  its  =  (3x3x3 

three  equal  factors.     We   first  resolve  2x3=6 

216  into  its  prime  factors,  finding  216=2x2x2x3x3x3. 
Since  there  are  three  2's,  one  of  the  three  equal  factors  will 
contain  2 ;  and  since  there  are  three  3's,  one  of  the  three  equal 
factors  will  contain  3 ;  hence  one  of  the  three  equal  factors  is 
2x3,  or  6. 

Case  III.  To  resolve  a  number  into  factors  bearing  a  cer- 
tain relation  to  each  other  In  this  case  we  may  divide  the 
given  number  by  the  product  of  the  numbers  representing  the 
relation  of  the  other  factors  to  the  smallest  factor,  then  resolve 
the  quotient  into  equal  factors,  and  then  multiply  this  equal 
factor  by  the  numbers  indicating  the  relation  of  the  other  fac- 
tors to  it. 

Thus,  resolve  384  into  three  factors,  such  that  the  second 

shall    be  twice  the  first  and   the  third   three  times  the  first. 

Since  the  second  factor  equals  2  times  the 

first  and  the  third  equals  3  times  the  first,    6)384 

the  product  of  the  factors  will  equal  2x3,         64=4  x  4x4 

or  6  times  the  first  factor,  used  three  times ;  „  x  7~r„ 

3  X  d —  1 Q 
hence  if  we  divide  384  by  6,  the  quotient, 

64,  will  be  the  product  of  the  smallest  factor  used  three  times  ; 

therefore,  if  we  resolve  64  into  three  equal  factors,  one  of  these 

factors  will  be  the  smallest  of  the  three  factors  required.     One 

of  the  three  equal  factors  of  64,  found  by  the  previous  case,  is 

4 ;  hence,  the  smallest  factor  is  4,  the  second  is  4x  2  or  8,  and 

the  third  is  4x3  or  12. 

Case  IY.    To  find  the    divisors  common  to  two  or  mo^e 


FACTORING.  247 

numbers  In  tbis  case  we  resolve  the  numbers  into  their 
prime  factors,  and  the  common  prime  factors  and  all  the  num- 
bers which  we  can  form  by  combining  them  will  be  all  the 
common  divisors. 

Thus,  find  the  divisors  common  operation. 

to    108   and   144.     Resolving   the  108=22x33 

numbers  into  their  prime  factors,  144=24x32 

we  find  the  common  factors  to  be  °om'  *actor=22x3' 

22x32;    hence,  1,  2,   4,  3,   9,  and     x  2  4 
all   the   possible   products    arising     1  3  9  2  6  18  4  12  36 
from  their  combination,  will  be  all 
the  divisors  of  108  and  144. 

Case  Y.  To  find  all  the  factors  or  divisors  of  a  number 
In  this  case  we  resolve  the  number  into  its  prime  factors,  form 
a  series  consisting  of  1  and  the  successive  powers  of  one  fac- 
tor, and  under  this  write  1  and  the  successive  powers  of  an- 
other factor,  and  take  the  products  of  the  terms  of  this  series, 
etc.     Thus,  find  all  the  different  divisors  of  108. 

The    factors   of    108 
are  two  2's  and  three  operation. 

3's.    Since  3  is  a  factor     J0^2  X2  X  3  x  3  x  3 
3  times,  1,  3,   32,  33,  is     x  2  4 

the  first  series  of  divis-     1  3  9  27  2  6  18  54  4  12  36  108 
ors;   and  since   2  is  a 

factor  twice,  1,  2,  22  is  the  second  series  of  divisors;  and  the 
products  of  the  terms  of  these  two  series  will  give  the  prime 
factors  and  all  possible  products  of  them ;  and  therefore,  all  the 
divisors  of  the  number. 

Case  YI.  To  find  the  number  of  divisors  of  a  number. 
In  this  case  we  resolve  the  number  into  its  prime  factors,  in- 
crease the  number  of  times  each  factor  is  used  by  1,  and  take 
the  product  of  the  results.  Thus,  find  the  number  of  divisors 
of  108. 


248  THE    PHILOSOPHY    OF    ARITHMETIC. 

Factoring,    we     find    108     equals  operation. 

22x33.    Now   it  is   evident    that    1  108=22x3s 

with  the  first  and  second  powers  of     (2+l)x(34-l)=12 
2  will  give  a  series  of  three  divisors;  and  1  with  the  first, 
second  and  third  powers  of  3,  will  give  a  series  of  four  divis- 
ors; hence  their  products  will   give  a  series  of  three  times 
four,  or  12  divisors. 


CHAPTER  IV. 

THE  GREATEST   COMMON   DIVISOR. 

A  DIVISOR  of  a  number  is  a  number  which  will  exactly 
divide  it.  A  number  is  said  to  exactly  divide  another 
when  it  is  contained  in  it  a  whole  number  of  times  without  a 
remainder.  A  Common  Divisor  of  two  or  more  numbers  is  a 
divisor  common  to  all  of  them.  The  Greatest  Common  Divi- 
sor of  several  numbers  is  the  greatest  divisor  common  to  all 
of  them.  By  using  the  word  factor  to  denote  an  exact  integral 
divisor,  we  may  define  as  follows: 

A  Divisor  of  a  number  is  a  factor  of  the  number.  A  Com- 
mon Divisor  of  two  or  more  numbers  is  a  factor  common  to 
all  of  them.  The  Greatest  Common  Divisor  of  several  num- 
bers is  the  greatest  factor  common  to  all  of  them.  These  defi- 
nitions employ  the  term  factor  with  a  derivative  signification. 
A  factor  is  primarily  one  of  the  makers  of  a  number,  entering 
into  its  composition  multiplicatively.  From  this  it  follows, 
however,  that  a  factor  is  an  integral  divisor  of  a  number,  and 
as  such,  it  may  be  conveniently  and  legitimately  used  in  defin- 
ing a  common  divisor. 

In  the  subject  of  greatest  common  divisor,  the  term  "  divisor" 
is  used  in  a  sense  somewhat  special.     It  signifies  an  exact 

divisor a  number  which   is  contained  a  whole  number  of 

times  without  a  remainder.  The  word  measure  was  formerly 
used  instead  of  divisor,  and  is  in  some  respects  preferable  to 
divisor.  A  common  divisor  of  several  numbers  is  appropri- 
ately called  their  common  measure,  since  it  is  a  common  unit 

(249) 


250  THE    PHILOSOPHY    OF   ARITHMETIC. 

of  measure  of  those  numbers.  The  term  measure,  in  this  sense, 
originated  in  Geometry,  where  a  line,  surface,  or  volume  which 
is  contained  in  a  given  line,  surface,  or  volume,  is  called  the 
unit  of  measure  of  the  quantity.  In  arithmetic,  the  term 
divisor  is  generally  preferred. 

Gases. — There  are  two  general  cases  of  greatest  common 
divisor,  growing  out  of  a  difference  in  the  method  of  treatment 
adapted  to  the  problems.  When  numbers  are  readily  factored, 
we  employ  one  method  of  operation  ;  when  they  are  not  readily 
factored,  we  are  obliged  to  employ  another  method.  This  dual 
division  of  the  subject  into  two  cases  is  thus  seen  to  be  founded, 
not  upon  any  distinctions  in  the  idea  of  the  subject,  but  upon 
the  method  of  operation  adapted  to  the  numbers  given.  These 
two  cases  are  formally  stated  as  follows : 

I.  To  find  the  greatest  common  divisor  when  the  numbers 
are  readily  factored. 

II.  To  find  the  greatest  common  divisor  when  the  numbers 
are  not  readily  factored. 

Treatment. — The  general  method  of  treatment  in  the  first 
case  is  to  analyze  the  numbers  into  their  factors,  and  take  the 
product  of  the  common  factors.  In  the  second  case  the  num- 
bers are  operated  upon  in  such  a  manner  as  to  remove  all  the 
factors  not  common,  and  thus  cause  the  greatest  common  divi- 
sor to  appear.  These  two  methods  will  be  made  clear  by  their 
application. 

Case  I.  To  find  the  greatest  common  divisor  when  the 
numbers  are  readily  factored. 

This  case  may  be  solved  by  two  distinct  methods.  The  first 
method  consists  in  writing  the  numbers  one  beside  another, 
and  finding  all  their  common  factors  by  division,  and  then  tak- 
ing the  product  of  these  common  factors.  To  illustrate,  re- 
quired the  greatest  common  divisor  of  42,  84,  and  126. 

1st  Method. — We  place  the  numbers  one  beside  another 
as  in  the  margin.  Dividing  by  2,  we  see  that  2  is  a  common 
factor  of  the  numbers.     Dividing  the  quotients  by  3,  we  see 


THE   GREATEST    COMMON   DIVISOR.  251 

that  3  is  a  common   factor  of  the  operation. 

numbers.  Dividing  these  quotients  2)42  84  126 

by  7,  we   see  that  7  is  a  common  3)21  42  63 

factor  of  the  numbers ;   and  since  7)7  14  21 

the  final  quotients  1,  2,  and  3   are  12     3 

prime  to  each  other,  2,  3,  and  7  are    GL  C.  D.=2x  3x7=42 
all  the  common  factors  of  the  given  numbers.    Hence  2x3x7 
or  42,  is  the  greatest  common  divisor  required.     This  method, 
so  far  as  I  can  learn,  was   published  first  by  the  author  of 
this  work,  in  1855.     It  is  now  in  several  different  text-books. 

The  second  method  consists  in  resolving  the  numbers  into 
their  prime  factors,  and  taking  the  product  of  all  the  common 
factors.  To  illustrate,  take  the  problem  already  solved  by  the 
first  method. 

2d  Method. — Resolving  the  num-  operation. 

bers   into  their  prime  factors,  we  42=2  x  3  x  7 

find  that  2,  3,  and  7,  are  factors  84=2x2x3x7 

126=2x3x3x7 
common    to   the    three    numbers;      ~    n   T)  =2x3x7=42 
hence  their  product,  which  is  42,  is 

a  common  divisor  of  the  numbers;  and,  since  these  are  all 
the  common  factors,  42  is  the  greatest  common  divisor. 

Case  II.  To  find  the  greatest  common  divisor  when  the 
numbers  are  not  readily  factored.  The  second  case  may  be 
solved  by  a  process  which  may  be  entitled  the  method  of  suc- 
cessive division.  It  consists  in  dividing  the  greater  number 
by  the  less,  the  less  number  by  the  remainder,  etc.,  until  the 
division  terminates,  the  last  divisor  being  the  greatest  common 
divisor.  To  illustrate,  suppose  it  be  required  to  find  the  great 
est  common  divisor  of  32  and  56.  operation. 

Method. — We  first  divide  56  by  32,  then       32)56(1 
divide  the  divisor,  32,  by  the  remainder,  32 

24;    then  divide   the  divisor,  24,  by  the  24)32(1 

remainder,  8,  and  find  there  is  no  remain-  24^ 

der;  then  is  8  the  greatest  common  divi-  8)24(3 

sor  of  32  and  56.  2— 


252  THE   PHILOSOPHY   OF    ARITHMETIC. 

This  method  is  applicable  to  all  numbers,  and  may  therefore 
be  distinguished  from  the  methods  of  the  previous  case  by 
naming    it    the    general    method,   those    being 
adapted  to  only  a  special  case.    A  more  conveni-     operation. 
ent  method  of  expressing  the  successive  divis-         24  32  1 
ion,  and  one  which   I   recommend   for   general         "gToTiq 
adoption,  is  that  represented  in  the  margin.     It  24 

is  observed  in  this  method  that  the  quotients 
are  all  written  in  one  column  at  the  right,  and  that  the  num- 
bers in  the  other  columns  become  divisors  and  dividends  in 
turn. 

Explanation. — In  the  explanation  of  the  rationale  of  the 
general  method  of  successive  division,  there  are  two  distinct 
conceptions  of  the  nature  of  the  process.  These  two  methods 
may,  for  convenience  in  this  discussion,  be  entitled  the  Old  and 
the  New  methods  of  explanation.  By  the  Old  Method  of 
explanation  I  mean  the  one  generally  given  in  the  text-books 
on  arithmetic  and  algebra.  The  New  Method  is  the  one  which 
is  found  in  my  own  mathematical  works.  I  will  present  each, 
pointing  out  the  difference  between  them.  Both  methods  are 
based  upon  the  following  general  principles  of  common 
divisor : 

1.  A  divisor  of  a  number  is  a  divisor  of  any  multiple  of 
that  number. 

2.  A  common  divisor  of  two  numbers  is  a  divisor  of  their 
sum,  and  also  of  their  difference. 

The  Old  Method  of  explaining  the  process  of  successive 
division  is  briefly  stated  in  the  following  propositions: 

1.  Any  remainder  which  exactly  divides  the  previous  divi- 
sor, is  a  common  divisor  of  the  two  given  quantities. 

2.  The  greatest  common  divisor  will  divide  each  remainder, 
and  cannot  be  greater  than  any  remainder. 

3.  Therefore,  any  remainder  which  exactly  divides  the 
previous  divisor  is  the  greatest  common  divisor. 

Whatever  the  special  form  of  the  old  method  of  explanation, 


THE   GREATEST   COMMON   DIVISOR.  253 

and  we  find  it  considerably  varied  by  different  authors,   it 
involves,  more  or  less  distinctly,  the  principles  just  stated 

The  New  Method  of  conceiving  of  the  nature  of  the  process 
and  explaining  it,  may  be  presented  in  the  following  princi- 
ples: 

1.  Each  remainder  is  a  number  op  times  the  greatest  com- 
mon divisor. 

2.  A  remainder  cannot  exactly  divide  the  previous  divisor 
unless  such  remainder  is  once  the  greatest  common  divisor. 

3.  Hence,  the  remainder  which  exactly  divides  the  previous 
divisor,  is  once  the  greatest  common  divisor. 

The  first  of  these  principles  is  evident  from  the  considera- 
tion that  a  number  of  times  the  greatest  common  divisor,  sub- 
tracted from  another  number  of  times  the  greatest  common 
divisor,  leaves  a  number  of  times  the  greatest  common  divisor. 

The  second  of  these  principles  becomes  evident  from  the 
consideration  that  of  any  remainder  and  the  previous  divisor, 
the  numbers  denoting  how  many  times  the  greatest  common 
divisor  is  contained  in  each  are  prime  to  each  other;  hence, 
one  cannot  divide  the  other  unless  one  of  these  numbers  is  a 
unit,  or  the  remainder  becomes  once  the  greatest  common 
divisor. 

These  principles  may  be  readily  seen  by  factoring  the  two 
numbers  and  then  dividing.     Thus,  in  the  problem   already 
given,  knowing   the  greatest  com- 
mon divisor  to  be  8,  we  may  re-  rtx„0P^fTI0N" 
solve  32  and  56  into  a  number  of        s     4x8 
times  8,  and  then  divide.     Observ-  3x8)4x8(1 
ing  the  operations  in  this  factored  '     3x8 
form,  we  see  that  each  remainder  1x8)3  X  8(3 
is  a  number  of  times  the  greatest  3x8 
common  divisor,  and  that  the  fac- 
tors 7  and  4,  and  also  4  and  3,  are  respectively  prime  to  each 
other;  and  also  that  the  division  terminates  when  we  reach  a 
divisor  which  is  once  the  greatest  common  divisor,  and  that  it 


254  THE    PHILOSOPHY   OF    ARITHMETIC. 

cannot  terminate  until  we  come  to  once  the  greatest  common 
divisor. 

In  arithmetic  I  find  it  simpler  to  pre-         operation. 
sent     this    New    Method,    in   a    manner       32)^C1 
slightly  varied  from  the  above,  preserving  — ^ 

its  spirit,  but  slightly  changing  the  form  ^v 

to  adapt  it  more  fully  to  the  comprehen-  — - 

sion  of  younger  minds.     To  illustrate,  let  24 

it  be  required  to  find  the  greatest  common  — - 

divisor  of  32  and  56.  Dividing  as  previously  explained,  we 
have  the  work  in  the  margin.  The  explanation,  showing  that 
this  process  will  give  the  greatest  common  divisor,  is  as  fol- 
lows: 

1st.  The  last  remainder,  8,  is  a  number  of  times  the  great- 
est common  divisor.  For,  since  32  and  56  are  each  a  number 
of  times  the  G.  C.  D.,  their  difference,  24,  is  a  number  of  times 
the  G.  C.  D.;  and  since  24  and  32  are  each  a  number  of  times 
the  G-.  C.  D.,  their  difference,  8,  is  also  a  number  of  times  the 
G.  C.  D. 

2d.  The  last  remainder,  8,  is  once  the  greatest  common 
divisor.  For,  since  8  divides  24,  it  will  divide  24  +  8,  or  32; 
and  since  it  divides  32  and  24,  it  will  divide  24+32,  or  56; 
and  now  since  8  divides  32  and  56,  and  is  a  number  of  times 
the  G.  C.  D.,  and  since  once  the  G.  C.  D.  is  the  greatest  num- 
ber that  will  divide  32  and  56,  therefore  8  is  once  the  G.  C.  D. 

This  second  method  of  conceiving  the  subject  is  believed  to 
be  the  true  one.  It  is  simpler  than  the  old  method,  and 
reaches  the  root  of  the  matter,  which  the  other  does  not.  It 
looks  down  into  the  process  and  sees  the  nature  of  the  remain- 
ders, and  their  relation  to  each  other.  All  the  remainders  are 
seen  to  be  a  number  of  times  the  greatest  common  divisor, 
each  being  a  less  and  less  number  of  times  the  greatest  com- 
mon divisor;  and  consequently,  if  the  division  be  continued  far 
enough,  we  will  at  length  arrive  at  once  the  greatest  common 
divisor.     The  object  of  dividing  is  thus  seen  to  be  a  search  for 


THE   GREATEST    COMMON   DIVISOR.  255 

a  smaller  number  of  times  the  greatest  common  divisor,  know- 
ing that  eventually  we  will  arrive  at  once  this  factor,  which  will 
be  indicated  by  the  termination  of  the  division.  The  experience 
of  the  class-room,  especially  in  the  sudden  revelation  of  the 
philosophy  of  the  division  to  those  who  thought  they  had  a 
clear  idea  of  the  subject  by  the  old  method,  has  frequently 
demonstrated  the  superiority  of  the  method  now  suggested. 
It  is  also  readily  seen,  from  this  conception  of  the  subject,  that 
the  secret  of  the  method  of  finding  the  greatest  common  divi- 
sor is  not  in  the  division  of  the  numbers,  but  in  the  subtrac- 
tion of  them — knowing  that  when  we  subtract  one  number  of 
times  a  factor  from  another  number  of  times  the  factor,  the 
remainder  is  a  less  number  of  times  the  factor,  and  that  the 
object  is  to  continue  the  subtraction  until  we  reach  once  the 
required  factor. 

Abbreviation. — This  view  of  the  subject  leads  us  to  discover 
a  shorter  process  of  obtaining  the  greatest  common  divisor 
than  that  of  the  ordinary  method  of  dividing. 
Thus,  suppose  we  wish  to  find  the  greatest  operation. 
common  divisor  of  32  and  116.  If  we  divide  in  32 
the  ordinary  way,  we  will  find  that  it  requires  five  36 
divisions  and  five  quotients.  If  we  take  4  times  4 
32  and  subtract  116  from  it,  we  get  a  smaller  re- 
mainder than  if  we  subtract  3  times  32  from  116, 
and  hence  are  nearer  once  the  greatest  common  divisor.  If  we 
then  subtract  32  from  3  times  12,  we  obtain  a  smaller  remainder 
than  if  we  subtract  2  times  12  from  32,  and  hence  are  nearer 
once  the  greatest  common  divisor,  etc.  This  latter  method 
requires  but  three  multiplications  and  subtractions,  and  hence 
saves  two-fifths  of  the  work.  In  many  problems  nearly  one- 
half  the  labor  is  saved  by  this  method. 

The  method  of  conceiving  and  explaining  the  greatest  com- 
mon divisor  here  given,  is  perhaps  most  clearly  exhibited  by 
the  use  of  general  symbols.  Thus,  let  A  and  B  be  any  two 
numbers,  of  which  A  is  the  greater ;  let  c  be  their  greatest 


116  4 
128 

12  3 
123 


256  THE   PHILOSOPHY    OF    ARITHMETIC. 

common  divisor,  and  suppose  A=ac  and  B= be;  then  dividing 

the  greater  by  the  less,  the  smaller  by  the  remainder,  and  thus 

continuing,  we  have  the  operation  in 

the  margin,  which  may  be  explained     b.c.)  a.  c(q 

as  follows:  j±—        __ 

1st.  Each  remainder  is  a  number  ^a~  <l)c—r-c 

of  times  the  G.  G.  D.    This  is  shown     r.c)  b.  c(q' 
by  the  division,  since  each  remainder  r(i  '  c 

is  a  number  of  times  c,  the  first  being  (o—rq  )c—r  .c 

{a—bq)  times  c,  which  we  indicate     r'-c)  r-  c(<?" 
by  r  times  c,  etc.  r  q  .c 

(r—r^Q^}c — r^  c 
2d.  A   remainder   cannot  exactly  v  *  ' 

divide  the  previous   divisor  unless 

such  remainder  is  once  the  G.  G.  D.  To  prove  this  it  must 
be  shown  that  b  and  r  are  prime  to  each  other;  also,  that  r  and 
r1  are  prime  to  each  other,  etc.  Now,  if  b  and  r  are  not  prime 
to  each  other,  they  have  a  common  factor,  and  hence,  r+bq  or 
a  contains  this  factor  of  b ;  but  a  and  b  are  prime  to  each 
other,  since  c  is  the  greatest  common  factor  of  a  and  b;  there- 
fore, b  and  r  are  prime  to  each  other.  In  the  same  way  it  may 
be  shown  that  r  and  r'  are  prime  to  each  other,  r'  and  r",  etc. 
Hence,  since  of  two  numbers  prime  to  each  other  one  cannot 
contain  the  other  unless  the  latter  is  a  unit,  a  remainder  can- 
not exactly  divide  the  previous  divisor  unless  such  remainder 
is  once  the  G.  C.  D. 

3d.  Hence,  the  remainder  which  does  exactly  divide  the  pre- 
vious divisor  is  once  the  Greatest  Common  Divisor. 


CHAPTER  V. 

THE  LEAST   COMMON  MULTIPLE. 

A  MULTIPLE  of  a  number  is  one  or  more  times  the  num- 
ber. A  Common  Multiple  of  two  or  more  numbers  is  a 
number  which  is  a  multiple  of  each  of  them.  The  Least 
Common  Multiple  of  several  numbers  is  the  least  number 
which  is  a  multiple  of  each  of  them. 

This  conception  of  a  multiple  is  that  it  is  a  number  of  times 
some  number.  It  regards  the  subject  as  a  special  case  of  form- 
ing composite  numbers.  A  common  multiple  is  a  synthesis  of 
all  the  different  factors  of  two  or  more  numbers,  giving  rise  to 
a  number  which  is  one  or  more  times  each  of  those  numbers. 
The  relation  of  the  subject  to  multiplication  is  also  seen  in  the 
term  multiple  itself.  The  primary  idea  is,  what  number  is  one 
or  more  times  each  of  several  numbers  ? 

This  view  of  a  multiple  differs  from  that  usually  presented 
by  our  writers  of  text-books.  The  usual  definition  is — A  mul- 
tiple of  a  number  is  a  number  which  exactly  contains  it.  This 
puts  containing  as  the  primary  idea,  and  makes  the  subject 
seem  to  originate  in  division  rather  than  in  multiplication. 
Indeed,  some  have  gone  so  far  in  this  direction  as  to  change 
the  name  from  multiple  to  dividend,  calling  it  a  common  divi- 
dend instead  of  a  common  multiple.  That  this  idea  is  incor- 
rect is  evident  both  from  the  term  multiple,  and  the  nature  of 
the  subject.  There  can  be  no  question  of  the  subject  having 
its  origin  in  multiplication,  and  it  should  certainly  be  defined 
In  accordance  with  this  view. 

17  (257) 


258  THE   PHILOSOPHY    OP   ARITHMETIC. 

It  will  be  observed  that  the  subject  of  Greatest  Common 
Divisor  is  placed  before  that  of  Least  Common  Multiple;  that 
is,  a  special  case  of  Factoring  before  a  special  case  of  Compo- 
sition, thus  reversing  the  general  order  of  synthesis  before 
analysis.  The  reason  for  this  is  that  Common  Multiple  is  a 
synthesis  of  factors,  and  in  some  numbers  these  factors  are 
most  conveniently  found  by  the  method  of  greatest  common 
divisor.  This  order  is  thus  a  matter  of  convenience  in  per- 
forming the  operation,  and  not  that  of  logical  relation. 

Cases. — There  are  two  general  cases  of  Least  Common 
Multiple,  as  of  Greatest  Common  Divisor.  This  distinction  of 
sases,  as  in  the  corresponding  analytic  process,  is  not  founded 
in  a  variation  of  the  general  idea,  but  rather  in  the  practical 
ease  or  difficulty  of  finding  the  factors  of  the  numbers.  When 
the  numbers  are  readily  factored  we  employ  one  method  of 
operation ;  when  they  are  not  easily  factored  we  employ  an- 
other method.    These  two  cases  are  formally  stated  as  follows : 

I.  To  find  the  least  common  multiple  when  the  numbers  are 
readily  factored. 

II.  To  find  the  least  common  multiple  when  the  numbers 
are  not  readily  factored. 

Treatment. — The  general  method  of  treatment  in  the  first 
case  is  to  resolve  the  numbers  into  their  different  factors 
by  the  ordinary  method  of  factoring,  and  take  the  product  of 
all  the  different  factors.  In  the  second  case,  the  different  fac- 
tors are  found  by  the  process  of  determining  the  greatest  com- 
mon divisor,  and  are  then  combined  as  before. 

Case  I.  To  find  the  least  common  multiple  when  the  num- 
bers are  readily  factored.  This  case  may  be  solved  by  two 
distinct  methods.  The  first  method  consists  in  resolving  the 
numbers  into  their  prime  factors,  and  then  taking  the  product 
of  all  the  different  prime  factors,  using  each  factor  the  greatest 
number  of  times  it  appears  in  either  number.  Thus,  required 
the  least  common  multiple  of  20,  30,  and  70. 

We    first    resolve    the   numbers  into  their   prime   factors 


THE   LEAST    COMMON  MULTIPLE.  259 

Since  the  factors  of  20  are  2x2x5,  the  multiple  must  con 

tain  the  factors  2,  2,  and  5  ; 

since  the  factors  of  30  are  o^o^o™'* 

„  „        ,  „     .  ,  .  20=2x2x5 

2,  3,  and  5,   it  must  contain  30=2x3x5 

the  factors  2,  3,  and  5;  and  70=2x5x7 

for  a  similar  reason  it  must     L.C.  M.  =  2x  2x3x5 x  7=420. 

contain  the  factors  2,  5,  and 

7 ;  hence,  the  least  common  multiple  of  20,  30,  and  70  must 

contain   the   factors  2,    2,    3,    5,   and   7,    and  no  others;  and 

their  product,   which   is   420,    is   the   least  common  multiple 

required. 

The  second  method  consists  in  writing  the  numbers  one 
beside  another  and  finding  all  the  different  factors  by  division, 
and  then  taking  the  product  of  these  factors.  To  illustrate, 
find  the  least  common  multiple  of  24,  30,  and  70. 

Placing  the  numbers  beside  one  another,  and  dividing  by  2, 
we  find  that  2  is  a  factor  of  all  the  numbers  ;  it  is  therefore  a 
factor  of  the  least  common  multiple.     Divid- 
ing the  quotients  by  3,  we  see  that  3  is  a  factor     operation. 

21^4^0   70 
of  some  of  the  numbers;  it  is  therefore  a  factor       H 

of  the  least  common  multiple.     Continuing  to       ' 
divide,  we  find  all  the   different  factors  of   the     5)_4_5_35 
numbers  to  be  2,  3,  4,  5,  and  7 ;  hence,  their  pro-  4     17 

duct,  which  is  840,  will  be  the  least  common  multiple  required. 
Case  II.  To  find  the  least  common  multiple  when  the  num- 
bers are  not  readily  factored.  The  second  case  is  solved  by  a 
method  which  may  be  called  the  method  of  greatest  common 
divisor.  By  it,  when  there  are  two  numbers,  we  find  the 
greatest  common  divisor  of  the  two  numbers  and  multiply  one 
of  them  by  the  quotient  of  the  other  divided  by  their  greatest 
common  divisor.  When  there  are  more  than  two  numbers, 
we  find  the  least  common  multiple  of  two  of  the  numbers,  and 
then  of  this  multiple  and  the  third  number,  etc.  To  illustrate, 
required  the  least  common  multiple  of  187  and  221. 


260  THE   PHILOSOPHY   OF   ARITHMETIC. 

We  first  find  the  greatest  common  divisor  to  be  IT.     Now, 
the   least    common    multiple   of  operation. 

187  and  221  must  be  composed  1871221 

of  all  the  factors  of  187,  and  all  170  187 


the  factors  of  221  not  contained  17  I  34 

in   187.      If  we  divide  221   by  I  jfi 

the  greatest  common  divisor,  we     ^  ^  =i87x2—  =2431 

shall  obtain  the  factors  of  221  not  1? 

belonging  to  187  ;  hence,  the  least  common  multiple  is  equal 

to  187x221-r-17,  which  we  find  is  2431. 

Another  statement  for  this  method  is,  divide  the  product  of 
the  two  numbers  by  their  greatest  common  divisor.  The  value 
of  this  method  may  be  seen  by  attempting  to  find  the  least 
common  multiple  of  1127053  and  2264159  by  each  method. 

This  method  is  very   clearly  ^_  ,  m_T 

,.,.,,  ,        -  «        .  OPERATION. 

exhibited  by  the  following  gen-  A=axc 

eral  explanation.     Let  A  and  B  B=bxc        A 

be  any  two  quantities,  and  let     L.  C.  M.=ax&Xc=  —  Xi? 

their  greatest  common  divisor  be 

represented  by  c,  and  the  other  factors  by  a  and  b,  respectively ; 

then  we  shall  have  the  L.  C.  M.=ax&Xc,  Case  L;  but  6Xc= 

A  A 

'  B,  and  o=  -  /  hence,  L.  C.  M.  =  -  X  B. 
c  c 


CHAPTER  VI. 

INVOLUTION. 

INVOLUTION  is  the  process  of  forming  composite  numbers 
by  the  synthesis  of  equal  factors.  It  is,  as  has  been  pre- 
viously explained,  a  special  case  of  Composition.  If  in  the 
general  synthesis  of  factors,  we  fix  upon  the  condition  that 
all  the  factors  are  to  be  equal,  the  process  is  called  Involution, 
and  the  composite  number  formed  is  called  a  Power  of  that 
factor. 

Involution  may,  therefore,  be  defined  as  the  process  of  rais- 
ing numbers  to  required  powers.  The  power  of  a  number  is 
the  product  obtained  by  using  the  number  as  a  factor  any  num- 
ber of  times.  The  different  powers  of  a  number  are  called, 
respectively,  the  square,  the  cube,  the  fourth  power,  etc.  The 
square  of  a  number  is  the  product  obtained  by  using  the  num- 
ber as  a  factor  twice.  The  cube  of  a  number  is  the  product 
obtained  by  using  the  number  as  a  factor  three  times.  These 
definitions,  which  are  beginning  to  be  adopted  by  authors,  are 
regarded  as  improvements  upon  those  framed  from  the  usual 
conception  of  the  subject. 

Symbol. — The  power  of  a  quantity  is  indicated  by  a  figure 
written  at  the  right,  and  a  little  above  the  quantity.  Thus, 
the  third  power  of  5  is  indicated  by  53.  The  earlier  writers  on 
mathematics  denoted  the  powers  of  numbers  by  an  abbrevia- 
tion of  the  name  of  the  power.  Harriot,  an  eminent  math- 
ematician of  the  16th  century,  repeated  the  quantity  to  indi- 
cate the  power ;  thus,  for  a  fourth  power  he  wrote  aaaa.  The 
present  convenient  system  of  exponents  was  introduced  by 

(261) 


262  THE    PHILOSOPHY    OF    ARITHMETIC. 

Descartes,  an  eminent  philosopher  and  mathematician  cele- 
brated for  his  "  cogilo,  ergo  sum,"  and  the  invention  of  the 
method  of  Analytical  Geometry. 

Cases. — To  raise  a  number  to  each  different  power  is  a  vari- 
ation of  the  general  idea,  and  might  be  regarded  as  presenting 
distinct  cases;  but  the  methods  of  operation  in  each  one  of  these 
cases  are  so  similar,  that  they  may  all  be  considered  under 
one  head.  In  raising  a  number  to  a  given  power,  we  may 
have  two  objects  in  view: — first,  merely  to  find  the  required 
power;  and  second,  to  ascertain  the  law  by  which  the  different 
parts  of  a  number,  as  expressed  in  the  Arabic  system,  are 
involved.  These  two  objects  require  different  methods  of  pro- 
cedure, and  upon  this  difference  of  method  we  may  found 
two  distinct  cases  of  involution.  In  practice,  it  is  convenient 
to  divide  the  second  case  into  the  consideration  of  the  square 
and  the  cube,  thus  making  three  cases.  These  cases,  formally 
expressed,  are  as  follows: 

I.  To  raise  a  number  to  any  required  power. 

II.  To  raise  a  number  to  the  second  power,  and  ascertain  the 
»aw  by  which  the  power  is  formed. 

III.  To  raise  a  number  to  the  third  power,  and  ascertain  the 
law  by  which  the  power  is  formed. 

Treatment. — The  general  method  of  treatment  is  to  involve 
the  factors  by  multiplication.  In  the  first  case  a  variation 
occurs  for  the  purpose  of  abbreviation,  giving  two  methods. 
In  the  second  and  third  cases  the  number  is  resolved  into  parts 
and  involved  in  two  different  ways,  giving  also  two  distinct 
methods.  The  treatment  of  both  of  these  cases  will  now 
be  presented. 

Case  I.    To  raise  a  number  to  any  required     operation. 
power.     This  case  may  be  solved  by  forming  a  4 

product  by  using  the  number  as  a  factor  as  many  _Jl 

times  as  there  are  UDits  in  the  index  of  the  power.  ^" 

Thus,  to  fiud  the  third  power  of  4,  we  multiply  — 

4  by  4  giving  16,  and   then  multiply  16  by  4 


INVOLUTION.  2G3 

giving  64,  which  is  the  cube  of  4,  since  the  number  is  used  as 
a  factor  three  times. 

In  all  powers  higher  than  the  cube,  we  may  abbreviate  the 
process  by  taking  the  product  of  one  power  by  another.    Thus, 
in  finding  the  8th  power  of  2,  we  may  first  find 
the  square  of  2,  which  is  4,  then  multiply  4,     operation. 
the    square,    by    itself,    obtaining    16,    the    4tb  9 

power  of  2,  and  then  multiply  16,  the  4th  power, 
by  itself,  giving  256,  the  8th  power  of  2.     This  4 

method  may  be  applied  to  all  powers  higher  T7T 

than  the  third,  and  is  much  more  convenient  in  \q 

practice.     Thus,  in  finding  the  5th  power,  we       "255 
may  take  the  product  of  the  2d  and  3d  powers, 
or  the  product  of  the  square  by  the  square  by  the  first  power ; 
in  finding  the  6th  power,  we  may  cube  the  2d  power,  or  square 
the  3d  power,  or  multiply  the  4th  power  by  the  square,  etc. 

Case  II.  Squaring  Numbers  and  finding  the  law.  This 
case  may  be  solved  by  two  distinct  methods.  The  first 
consists  in  separating  the  number  into  its  elements  of  units, 
tens,  etc.,  and  multiplying  as  in  algebra  so  as  to  exhibit  the 
law  by  which  the  parts  are  involved.  The  second  method  per- 
forms the  process  of  involution  as  determined  by  the  building 
up  of  a  figure  in  geometry.  These  two  methods  may  be  dis- 
tinguished as  the  algebraic  and  geometric,  or  the  analytic  and 
synthetic  methods.  The  ultimate  object  of  these  methods 
is  to  derive  a  law  of  involution  by  which  we  may  be  able  to 
derive  methods  of  evolution.  These  two  methods  apply  both  to 
the  squaring  and  cubing  of  numbers.  The  synthetic  method 
cannot  be  extended  beyond  the  cubing  of  numbers ;  the  analytic 
method  is  general  and  will  apply  to  all  powers,  but  is  of  no 
practical  use  in  arithmetic  beyond  the  cube.  We  will,  there- 
fore, apply  these  two  methods  only  to  the  squaring  and  cubing 
of  numbers. 

Analytic  Method. — By  the  Analytic  Method  of  squaring 
numbers,  we  separate  the  number  into  its  units,  tens,  etc ,  and 


264  THE    PHILOSOPHY    OF    ARITHMETIC. 

keep  these  elements  distinct  in  the  involution  of  the  number, 
so  that  the  law  of  the  process  becomes  apparent.  To  illustrate, 
find  the  square  of  25. 

Twenty-five  equals  20  +  5,  or  2  tens  operation. 

and  5  units.     Writing  the  number  as  20+5 

in  the   margin,  and  multiplying  by  5  20  +  5 

and  by  20,  and  taking  the  sum  of  these  ^x^+^ 

products,  we  have  202+2  (5x20)+52.    ^0+5x20 _ 

From  this  we  see  that  the  square  of  a      20  +2(5><20)+5a 
number  consisting  of  tens  and  units,  equals  the  tens*+2  times 
tens  x  units-\-units2. 

If  we  involve  in  the  same  manner  a  number  consisting  of 
hundreds,  tens,  and  units,  we  shall  find  the  following  law: 
The  square  of  a  number  consisting  of  hundreds,  tens,  and  units 
equals  hundreds2 +2  x  hundreds  x  tens+tens2i-2  x  {hundreds-^ 
tens)  x  units+units1. 

Synthetic  Method. — The  Synthetic  Method  of  solving  the 
same  problem  is  as  follows:  Let  the  line  AB  represent  a 
length  of  20  units,  and  BE,  5  units. 
Upon  AB  construct  a  square :  the 
area  will  be  202=400  square  units. 
On  the  two  sides  DC  and  BC  con- 
struct rectangles  each  20  units  long 
and  5  broad,  the  area  of  which  will  be 
5X20=100,  and  the  area  of  both  will 
be  2x100=200  square  units.  Now 
add   the   little    square  on    CGr,  whose 

area  is  52=25  square  units,  and  the  sum  of  the  different  areas, 
400+200+25=625,  is  the  area  of  a  square  whose  side  is 
25. 

When  there  are  three  figures,  after  completing  the  second 
square  as  above,  we  must  make  additions  to  it  as  we  did  to  the 
first  square.  When  there  are  four  figures  there  are  three  addi- 
tions, etc. 

Case  III.   Cubing  Numbers  to  find  the  law.      This  case 


c 

INVOLUTION. 


265 


may  also  be  solved  by  two  distinct  methods,  as  in  squaring 
numbers,  which  we  distinguish  as  the  analytic  and  synthetic 
methods.  The  former  involves  the  number  by  the  method  of 
algebra ;    the  latter  by  the  principles  of  geometry. 

Analytic  Method.— By  the  Analytic  Method  we  resolve  the 
number  into  its  elements  of  units,  tens,  etc.,  and  keep  it  in 
this  form  as  we  perform  the  process  of  involution,  that  we 
may  exhibit  the  law  by  which  the  elements  of  a  number  enter 
into  its  cube.     To  illustrate,  find  the  third  power  of  25. 

Resolving  the  number  operation. 

into   its   units    and    tens  252=202+2(5x  20) +  52 

and  squaring  as  above,  we __ Z 

have      20>+2(5x20)+5'.  ,     6xf0't2J?SX2S+6' 

Multiplying  the  square  by       _^!+^l^±^^ 
5   aud  then   by  20,   aud       20M-3x5x20'+3x5'x20+5* 

taking  the  sum  of  the  products,  we  have  the  cube  of  25,  as 
given  in  the  margin.  Examining  the  result,  we  see  that  the 
cube  of  a  number  of  two  digits  equals  tensz+3*tens2Xunits-\- 
3X  tensXunits2-\-unitsz. 

Cubing  a  number  of  three  digits,  we  obtain  the  following  law : 
The  cube  of  a  number  of  three  digits  equals  hundredsz+3X 
hundreds2  X  tens  +  3  X  hundreds  Xtens2Jrtensz+3X{hundreds+ 
tensyxunits+3x(hundreds+tens)Xunits2+units\ 

Synthetic  Method. — By  the  Synthetic  Method  we  use  a 
cube  to  determine  the  process  of  involution.  To  illustrate, let 
us  find  the  cube  of  45  by  this  method. 

Let  A,  Fig.  1,  represent  a  cube  whose  sides  are  40  units;  its 
contents  will  be  403=64000.  We  then  wish  to  increase  the 
size  of  this  cube  so  that  its  sides  will  be  45  units.  To  in- 
crease its  dimensions  by  5  units,  we  must  add  first  the  three 
rectangular  slabs,  B,  C,  D,  Fig.  2;  2d,  the  three  corner  pieces, 
E,  F,  G,  Fig.  3 ;  3d,  the  little  cube  H,  Fig.  4.  The  three  slabs 
B,  C,  D,  are  40  units  long  and  wide  and  5  units  thick;  hence, 
their' contents  are  402X5X3=24000;  the  contents  of  the  cor- 
ner pieces,  E,  F,  G,  Fig.  3,  whose  length  is  40  and  breadth 
12 


266 


THE    PHILOSOPHY    OF    ARITHMETIC. 


40X52X3=3000,  and    the   contents 


Fig.  1. 


Fig.  2. 


and  thickness  5,  equal 
of  the  little  cube 
H,  Fig.  4,  equal 
53=125;  hence  the 
contents  of  the 
cube  represented  by 
Fig.  4  are  64000+ 
24000  +  3000+125= 
91125.  Therefore, 
the  cube  of  45,  etc. 

Here  we  see  that 
403  is  the  cube  of 
the  tens;  402x5x3 
is  tens3xunitsxS\ 
40x52x3is3xte?is 
Xunits*;  and  53  is 

units3;  hence  we  have,  as  before,  the 
cube  of  a  number  of  tens  and  units 
equals  fens3+3  x  tens2  x  units+3  X  tens 
Xunits2-\-  units3. 

When  there  are  three  figures  in  the 
number,  we  complete  the  second  cube     Hence,  45  =91125 
as  above,  and  then  make  additions  and  complete  the  third  in 
the  same  manner.     If  there  are  still  some  figures,  and  no  more 
blocks  to  make  additions,  let  the  first  cube  represent  the  cube 
already  found,  and  then  proceed  as  at  first. 


OPERATION. 

403=64000 

402X  5x3=24000 

40x52x3=  3000 

53=    125 


CHAPTER  VII. 

EVOLUTION. 

EVOLUTION  is  the  process  of  finding  one  of  the  several 
equal  factors  of  a  number.  It  is  an  analytic  process,  the 
converse  of  the  process  of  Involution.  Involution  is  a  synthe- 
sis of  equal  factors  ;  Evolution  is  an  analysis  into  equal  factors. 
The  former  is  a  special  case  of  composition;  the  latter  is  a 
special  case  of  factoring.  One  finds  its  origin  in  multiplica- 
tion; the  other  in  division.  Both  are  contained  in  the  primary 
synthetic  and  analytic  ideas,  and  are  the  result  of  pushing  for- 
ward and  specializing  those  notions. 

Any  one  of  the  several  equal  factors  of  a  number  is  called 
a  root  of  that  number.  The  degree  of  a  root  depends  upon  the 
number  of  equal  factors.  The  square  root  of  a  number  is  one 
of  its  two  equal  factors.  The  cube  root  of  a  number  is  one  of 
its  three  equal  factors,  etc.  These  definitions  are  regarded  as 
an  improvement  upon  the  old  ones,  that  the  square  root  of  a 
number  is  a  number  which  multiplied  by  itself  will  produce 
the  number,  and  similarly  for  the  other  roots.  Evolution  may 
also  be  defined  as  the  process  of  finding  any  required  root  of 
a  number. 

Symbol. — The  Symbol  of  Evolution  is  v/,  called  the  radical 
sign.  This  sign  was  introduced  by  Stifelius,  a  German  math- 
ematician of  the  15th  century.  It  is  a  modification  of  the 
letter  r,  the  initial  of  radix,  or  root.  Formerly,  the  letter  r 
was  written  before  the  quantity  whose  root  was  to  be  extracted, 
and  this  gradually  assumed  its  present  form,  <J '. 

To  indicate  the  degree  of  the  root  to  be  extracted,  a  figure 

(267) 


268  THE    PHILOSOPHY    OF    ARITHMETIC. 

is  prefixed  to  the  radical  sign ;  thus,  <$/,  ■$/,  J/,  etc.,  denote 
respectively  the  square  root,  cube  root,  fourth  root,  etc.  This 
figure  is  called  the  index  of  the  root,  because  it  indicates  the 
root  required.  The  index  of  the  square  root  is  usually  omitted, 
perhaps  because  the  symbol  was  applied  to  the  square  root  for 
some  time  before  its  use  was  extended  to  the  higher  roots. 
The  roots  of  numbers  are  also  indicated  by  fractional  expo- 
nents; as  4^,  8*,  etc. 

Cases. — Each  different  root  might  be  regarded  as  constitut- 
ing a  distinct  case,  but  it  is  most  convenient  to  treat  the  sub- 
ject under  three  general  cases,  as  in  Involution.  These  three 
cases  correspond  to  those  of  Involution,  and  may  be  formally 
expressed  as  follows : 

I.  To  extract  any  root  of  a  number  when  it  can  be  conven- 
iently resolved  into  its  prime  factors. 

II.  To  extract  the  square  root  of  a  number  when  it  can  not 
be  conveniently  factored. 

III.  To  extract  the  cube  root  of  a  number  when  it  can  not 
be  conveniently  factored. 

Treatment. — The  general  method  of  treatment  is  to  analyze 
the  number  into  the  parts  required.  In  the  first  case,  we  ana- 
lyze the  number  into  its  prime  factors,  and  then  make  a  syn- 
thesis of  some  of  these  factors.  In  the  second  and  third  cases, 
we  separate  the  number  into  parts  by  several  distinct  methods, 
corresponding  to  those  of  Involution. 

Case  I.  To  extract  any  root  when  the  number  can  be  readily 
factored.  This  case  is  solved  by  resolving  the  number  into 
its  prime  factors,  and  then  involving  the  factors  so  as  to  obtain 
the  equal  factor  required.  For  the  square  root  we  take  the 
product  of  each  of  the  two  equal  factors ;  for  the  cube  root  we 
take  the  product  of  each  of  the  three  equal  factors,  etc. 

Thus,   to  find   the    square    root    of 
1225,  we  resolve  the  number   into  its  operation. 

prime   factors,  5,  5,  7,  7,  and   take   the      1|25"T5z^55  X^ol 
product  of  one  of  the  two  5's  and  one  of 
the  two  7's,  giving  us  5  x  7,  or  35. 


EVOLUTION.  269 

To  find  the  cube  root  of  1728  ^operation. 

we  resolve  the  number  into  its     q^^Iq^^^  4x4 
prime   factors,  as  shown  in  the 

margin,  and  take  the  product  of  one  of  the  three  3's,  and  one 
of  the  three  4's,  giving  3x4,  or  12.  In  a  similar  manner  we 
find  aDy  root  of  any  perfect  power  that  can  be  resolved  into 
its  prime  factors. 

Case  II.  To  extract  the  Square  Root  of  a  number.  The 
Square  Root  of  a  number  is  one  of  the  two  equal  factors  of  the 
number.  The  square  root  of  a  number  may  also  be  defined  to 
be  a  number  which,  used  as  a  factor  twice,  will  produce  the 
given  number.  The  former  definition  is  somewhat  analytic; 
the  process  of  thought  is  from  the  number  to  its  elements.  The 
latter  is  rather  synthetic  ;  the  process  of  thought  is  from  the 
elements  to  the  number. 

The  method  of  extracting  the  square  root  of  a  number  con- 
sists in  analyzing  the  number  into  two  equal  multiplicative 
parts.  This  is  done  by  first  finding  the  highest  term  of  the 
root,  taking  its  square  out  of  the  number,  and  using  it,  accord- 
ing to  the  laws  of  involution,  to  determine  the  next  term  of 
the  root,  etc.  The  method  being  found  in  all  the  works  on 
arithmetic,  need  not  be  stated  here. 

Explanation. — There  are  two  methods  of  deriving  the  rule 
for  square  root,  or  of  explaining  the  reason  for  the  operation. 
These  methods  are  distinguished  as  the  Analytic  and  Synthetic 
methods.  The  former  consists  in  resolving  the  number  into 
its  elements  by  the  laws  obtained  by  the  analytic  method  of 
involution ;  the  latter  consists  in  finding  the  root  by  means  of 
a  geometrical  diagram  by  reversing  the  process  of  the  corres- 
ponding method  of  involution.  The  synthetic  method  will 
apply  to  both  the  square  and  cube  root  of  numbers,  but  cannot 
be  extended  beyond  the  cube  root.  The  analytic  method  is 
general,  and  can  be  applied  to  the  determining  of  any  root  of 
a  number. 

In  order  to  determine  how  many  figures  there  are  in  the  root, 


270  THE    PHILOSOPHY    OF   ARITHMETIC. 

and  where  to  begin  the  extraction  of  the  root,  we  employ  the 
following  principles: 

1.  The  square  of  a  number  consists  of  twice  as  many  fig- 
ures as  the  number,  or  of  twice  as  many  less  one. 

This  principle  may  be  demonstrated  as  follows:  Any  integral 
number  between  1  and  10  consists  of  one  figure,  and  any  num- 
ber between  their  squares,  1  and  100,  con- 
sists of   one  or    two    figures;    hence    the  12=1 

square  of  a  number  of  one  figure  is  a  num-       ,  r;:,5^: nAn 
u  *  *        fl  a  i  1002=  10,000 

ber  of  one  or  two  figures.     Any  number     10002=1  000  000 

between  10  and  100  consists  of  two  figures, 

and   any  number   between   their  respective  squares,  100  and 

10,000,  consists  of  three  or  four  figures;  hence,  the  square  of 

a  number  of  two  figures  is  a  number  of  three  or  four  figures, 

etc.     Therefore,  etc. 

2.  If  a  number  be  pointed  off  into  periods  of  two  figures 
each,  beginning  at  units  place,  the  number  of  full  periods, 
together  with  the  partial  period  at  the  left,  if  there  be  one,  will 
equal  the  number  of  places  in  the  square  root. 

This  is  evident  from  Prin.  1,  since  the  square  of  a  number 
contains  twice  as  many  places  as  the  number,  or  twice  as  many 
less  one. 

Analytic  Method. — By  the  analytic  method  of  explaining 
the  process  of  extracting  the  square  root  of  numbers,  we  re- 
solve the  number  into  its  elements,  and  derive  the  method  of 
operation  by  knowing  the  law  of  the  synthesis  of  these  elements. 
It  is  appropriately  named  the  analytic  method,  because  it  ana- 
lyzes a  number  into  its  elements,  and  operates  by  reversing  the 
synthetic  process  of  involution.  We  will  illustrate  this  method 
by  extracting  the  square  root  of  625. 

Explanation. — By  the  principles  of  involution  we  see  that 
there  will  be  two  figures  in  the  root,  hence  the  number  con- 
sists of  the  square  of  the  tens  plus  the  units  of  the  root,  which 
equals  the  square  of  the  tens,  plus  twice  the  tens  into  the 
units,  plus  the  square  of  the  units.     The  greatest  number  of 


EVOLUTION.  271 

tens  whose  square  is  contained  in  625     t2+2lu+u2=  6*25(25 
is  2  tens  •  squaring  the  tens  and  sub-    <2  =  20J  =400 

tracting  we  have  225,  which  equals     2/u-f  w2        =225 

twice  the  tens  into  the  units,  plus  the  2£=40 

u==  5 
square  of   the  units.       Now,    since  2Jtt4-tt2=225 

twice  the  tens  into  the  units  is  usually 

much  greater  than  the  units  squared,  225  consists  principally 
of  twice  the  tens  into  the  units;  hence  if  we  divide  225  by 
twice  the  tens,  we  can  ascertain  the  units.  Twice  the  tens 
equals  20x2,  or  40;  dividing  225  by  40,  we  find  the  units  to 
be  5,  etc. 

In  the  margin  the  law  of  the  involution  of  the  elements  is 
shown  by  the  use  of  the  letters  t  and  u,  the  initials  of  tens  and 
units.  This  representation  of  the  law  of  the  formation  of  the 
number  enables  us  to  separate  it  into  its  elements. 

Synthetic  Method. — By  the  synthetic  method  we  use  a 
geometrical  figure  and  derive  the  process  from  the  method  of 
forming  a  square  whose  area  shall  equal  the  given  number.  It 
is  called  synthetic  because  we  commence  with  a  smaller  square 
and  add  parts  to  it,  until  we  find  a  square  of  the  required  area. 
The  method  of  forming  the  square  will  give  us  a  method  of 
finding  the  square  root.  To  illustrate,  let  it  be  required  to  ex- 
tract the  square  root  of  625. 

Explanation. — The  greatest  number  of  tens  whose  square 
is  contained  in  625  is  2  tens.  Let  A,  Fig.  1,  represent  a  square 
whose  sides  are  2  tens  or  20  units,  its  area 
will  be  the  square  of  20,  or  400.  Subtract- 
ing 400  from  625,  we  have  225,  hence  our 
square  is  not  large  enough  by  225 ;  we  must 
therefore  increase  it  by  225.  To  do  this  we 
add  the  two  rectangles  B  and  C,  each  of 
which  is  20  units  long,  and  since  they  near- 
ly complete  the  square,  their  area  must  be  nearly  225  units ; 
hence,  if  we  divide  by  their  length  we  can  find  their  width. 
Their  length  is  20x2=40,  hence  their  width  is  225-T-40  or  5. 


c 

A  B 


272  THE   PHILOSOPHY    OF   ARITHMETIC. 

Now  complete  the  square  by  the  addition  of  the  little  corner 
square  whose  side  is  5  units,  and  then  the  entire  length  of  all 
the  additions  is  40+5,  or  45  units,  and  multiplying  by  the 
width  we  find  their  area  to  be  225  square  units.  Subtracting, 
nothing  remains;  hence,  the  side  of  a  square  which  contains 
625  square  units  is  25  units. 

The  same  method  will  apply  when  there  are  more  than  two 
figures  in  the  root.  The  methods  of  operation  indicated  by 
both  the  analytic  and  synthetic  methods  of  explanation,  are 
the  same.  These  methods  give  the  usual  rule  for  the  extrac- 
tion of  the  square  root. 

Case  III.  The  Cube  Root  of  Numbers.  The  Cube  Root  of 
a  number  is  one  of  the  three  equal  factors  of  the  number. 
The  cube  root  of  a  number  may  also  be  defined  to  be  a  number 
which,  used  as  a  factor  three  times,  will  produce  the  given 
number.  Again,  the  cube  root  of  a  number  may  be  defined  as 
a  number  which,  raised  to  the  third  power,  will  produce  the 
given  number.  These  definitions  are  all  correct,  though  they 
differ  in  idea.  The  first  is  analytic ;  the  thought  is  from  the 
number  to  its  elements.  The  second  and  third  are  synthetic; 
the  process  of  thought  is  from  the  elements  to  the  number. 

The  method  of  extracting  the  cube  root  of  a  number  consists 
in  analyzing  it  and  finding  one  of  its  three  equal  multiplicative 
parts.  This  is  done  by  first  finding  the  highest  term  of  the 
root  and  taking  its  cube  out  of  the  number,  then  finding  the 
second  term  by  means  of  the  first  term,  taking  their  combiua- 
tion  out  of  the  number,  etc.  There  are  several  methods  of 
doing  this,  the  three  most  important  of  which  may  be  distin- 
guished as  the  Old  Method,  a  New  Method,  and  Horner's 
Method.  There  are  several  other  methods,  which  I  do  not 
regard  of  sufficient  importance  to  consider  in  this  work. 

Old  Method. — The  Old  Method  is  so  called  because  it  is  the 
one  which  has  for  a  long  time  been  taught  and  practiced.  It 
may  be  distinguished  by  the  use  of  300  and  30  in  finding  trial 
and  complete  divisors.    By  a  slight  modification  of  the  method 


EVOLUTION.  273 

the  ciphers  of  these  multipliers  may  be  omitted,  and  this  form 
of  the  method  is  now  generally  preferred.  The  method  may 
be  stated  as  follows: 

Rule. — I.  Separate  the  number  into  periods  of  three  figures 
each,  beginning  at  units  place. 

II.  Find  the  greatest  number  whose  cube  is  contained  in  the 
left-hand  period ;  place  it  at  the  right  and  subtract  its  cube 
from  the  period,  and  annex  the  next  period  to  the  remainder 
for  a  dividend. 

III.  Take  3  times  the  square  of  the  first  term  of  the  root 
regarded  as  tens  for  a  trial  divisor  ;  divide  the  dividend  by 
it,  and  place  the  quotient  as  the  second  term  of  the  root. 

IV.  Take  3  times  the  last  term  of  the  root  multiplied  by  the 
preceding  part  regarded  as  tens;  write  the  result  under  the 
trial  divisor,  and  under  this  write  the  square  of  the  last  term 
of  the  root ;  their  sum  will  be  the  complete  divisor. 

V.  Multiply  the  complete  divisor  by  the  last  term  of  the 
root;  subtract  the  product  from  the  dividend,  and  to  the 
remainder  annex  the  next  period  for  a  new  dividend.  Take 
3  times  the  square  of  the  root  now  found,  regarded  as  tens,  for 
a  trial  divisor,  and  find  the  third  term  of  the  root  as  before; 
and  thus  continue  until  all  the  periods  have  been  used. 

Explanation. — This  process  of  extracting  cube  root  may  be 
explained  by  two  distinct  methods,  distinguished  as  the  ana- 
lytic and  synthetic  methods.  The  analytic  method  consists  in 
resolving  the  number  into  its  elements  by  the  laws  obtained 
from  the  analytic  method  of  involution.  The  synthetic  method 
consists  in  ascertaining  the  different  terms  of  the  root  by  the 
building  up  of  a  geometrical  cube. 

In  order  to  determine  the  number  of  figures  in  the  root  and 
with  what  part  of  the  number  to  begin  the  evolution,  it  is 
necessary  to  state  and  demonstrate  the  following  principle : 

I.  The  cube  of  a  number  consists  of  three  times  as  many 
figures  as  the  number,  or  of  three  times  as  many  less  one  ot 
two. 

18 


274  THE    PHILOSOPHY    OF    ARITHMETIC. 

This  principle  may  be  demonstrated  as  follows:    Any  inte- 
gral number  between  1  and  10  consists  of  one  figure,  and  any 
integral   number  between  their  cubes,   1 
and  1000,  consists  of  one,  two,  or  three         1  =1 

figures;  hence  the  cube  of  a  number  of     -  „«„~  1  „„„  nAA 
&         '         .  ,  ,  100—1, 000,000 

one   figure   is  a  number  or    one,  hvo,  or 

three  figures.     Any  number  between  10  and  100  consists  of 

two  figures,  and  any  number  between  their  cubes,   1000  and 

1,000,000,  consists  of  four,  five,  or  six  figures  ;  hence  the  cube 

of  a  number  of  two  figures  consists  of  three  times  two  figures, 

or  three  times  two,  less  one  or  two  figures. 

2.  If  a  number  be  pointed  off  into  periods  of  three  figures 
each,  beginning  at  units  place,  the  number  of  full  periods 
together  with  the  partial  period  at  the  left,  if  there  be  one,  will 
equal  the  number  of  figures  in  the  root. 

This  is  evident  from  Prin.  1,  since  the  cube  of  a  number  con- 
tains three  times  as  many  places  as  the  number,  or  three  times 
as  many,  less  one  or  two. 

Analytic  Method. — By  the  analytic  method  of  explaining 
the  process  of  extracting  the  cube  root  of  numbers,  we  resolve 
the  number  into  its  elements  and  derive  the  process  by  knowing 
the  law  of  the  synthesis  of  these  elements  in  the  process  of 
involution.  We  will  illustrate  the  method  by  the  solution  of 
the  following  problem:    Required  the  cube  root  of  91125. 

Solution. — Since   the  cube   of 
a  number  consists  of  three  times  91 '125(40 


as  many  places  as  the  number 


403=64  000     5 


27125  45 


27125 


itself,  or  of  three  times  as  many  4^  x*    4800 

,  ,        .,  ,  .     Jf      40X5X3=  600 

less  one  or  two,  the  cube  root  of  ,2=     or 

91125  will  consist  of  two  places,  rj^ 

and   hence    consist  of  tens  and  — 

units,  and  the  given  number  will  consist  of  the  cube  of  the  tens, 

plus  three  times  the  square  of  the  tens  into  the  units,  plus  three 

times  the  tens  into  the  square  of  the  units,  plus  the  cube  of  the 

units 


EVOLUTION.  275 

The  greatest  number  of  tens  whose  cube  is  contained  in  the 
given  number  is  4  tens.  Cubing  the  tens  and  subtracting,  we 
have  27125,  which  equals  three  times  the  square  of  the  tens 
into  the  units,  etc.  Now,  since  three  times  the  square  of  the 
tens  into  the  units  is  much  greater  than  all  the  rest  of  the  ex- 
pression, 27125  must  consist  principally  of  three  times  the 
square  of  the  tens  into  the  units ;  hence  if  we  divide  by  three 
times  the  square  of  the  tens  we  can  ascertain  the  units.  Three 
times  the  tens  squared  equals  3X402=4800;  dividing  by 
4800  we  find  the  units  to  be  5.  We  then  find  three  times  the 
tens  into  the  units  equal  to  40x5x3=600,  and  units  squared 
equals  52=25.  Taking  the  sum  and  multiplying  by  the  units, 
we  have  27125,  and  subtracting,  nothing  remains.  Hence  the 
cube  root  of  91125  is  45.  From  this  solution  we  readily  derive 
the  rule  given  above. 

Synthetic  Method. — By  the  synthetic  method  of  explana- 
tion we  use  a  geometrical  figure,  a  cube,  and  derive  the  process 
from  the  method  of  forming  a  cube  whose  contents  shall  equal 
the  number  of  units  in  the  given  number.  The  number  is 
regarded  as  expressing  the  number  of  cubic  units  in  a  cubical 
block,  the  number  of  linear  units  in  whose  side  will  be  the 
cube  root  of  the  number.  It  is  appropriately  called  synthetic, 
since  we  begin  with  a  cube  and  add  parts  to  it  until  we  find  a 
cube  of  the  required  contents.  The  method  of  forming  the 
cube  indicates  the  process  of  finding  the  cube  root.  This 
method  may  be  illustrated  by  the  solution  of  the  problem 
already  given:  Required  the  cube  root  of  91125. 

Solution. — We  find  the  number  of  figures  in  the  root  as 
before,  and  then  proceed  as  follows :  The  greatest  number  of 
tens  whose  cube  is  contained  in  the  given  number  is  4  tens. 

Let  A,  Fig.  1,  represent  a  cube  whose  sides  are  40,  its  con- 
tents will  be  403=64000.  Subtracting  from  91125  we  find  a 
remainder  of  27125  cubic  units;  hence,  the  cube  A  is  not  large 
enough  to  contain  91125  cubic  units  by  27125  cubic  units;  we 
will  therefore  increase  it  by  27125  cubic  units. 


276 


THE    PHILOSOPHY    OF    ARITHMETIC. 


To  do  this  we  add  Fig.  1.  Fig.  2. 

the  three  rectangular 
slabs  B,  C,  D,  Fig. 
2,  each  of  which  is 
40  units  in  length 
and  breadth ;  and 
since  they  nearly 
complete  the  cube, 
their  contents  must 
be  nearly  27125; 
hence,  if  we  divide 
27125  by  the  sum  of 
the  areas  of  one  of 
their  faces  as  a  base, 
we  can  ascertain 
their  thickness. 

The  area  of  a  face  of  one  slab 
is  402=1600,  and  of  the  three, 
3  X  1600  =  4800;  and  dividing 
27125  by  4800  we  have  a  quo- 
tient of  5;  hence  the  thickness 
of  the  additions  is  5  units.  We 
now  add  the  three  corner  pieces 
E,  F,  and  G,  each  of  which  is  40  units  long,  5  wide,  and  5 
thick;  hence  the  surface  of  a  face  of  each  is  40X5=200  square 
units,  and  of  the  three  it  is  200X3=600  square  units. 

We  now  add  the  little  corner  cube  H,  Fig.  4,  whose  sides  are 
5  units,  and  the  surface  of  a  face  is  52=25.  We  now  take  the 
sum  of  the  surfaces  of  the  additions,  and  multiply  this  by  the 
common  thickness,  which  is  5,  and  we  have  their  solid  contents 
equal  to  (4800+600+25)X5=27125.  Subtracting,  nothing 
remains;  hence  the  cube  which  contains  91125  cubic  units  is 
40+5  or  45  units  on  a  side. 

When  there  are  more  than  two  figures  we  increase  the  size 
of  the  new  cube,  Fig.  4,  as  we  did  the  first,  or  let  the  first 
cube,  Fig.  1,  represent  the  new  cube,  and  proceed  as  before. 


OPERATION. 

91-125(40 
403=64  000  _5^ 

402X3=4800 

40X5X3=  600 

52=_25 

5425 


27125  45 


27125 


EVOLUTION.  277 

Infc  ftc*£zL04b  C^aiiAA^D. — These  two  methods  of  explain- 
ii-ft  the  process  ot  extracting  ihe  square  and  cube  roots  of  num- 
bers arc  entirely  distinct:  they  die  based  upon  different  ideas, 
though  they  give  rise  to  the  same  practical  operation.  The 
synthetic  method  is  the  one  generally  given  in  the  text-books 
on  arithmetic ;  the  analytic  method  was,  until  recently,  confined 
to  algebra.  It  has  been  a  question  which  of  these  methods  of 
explanation  is  the  better,  some  preferring  ihe  one  and  some  the 
other.  In  my  own  opinion  the  analytic  method  is  to  be  pre- 
ferred for  several  reasons,  among  which  the  following  may  be 
stated : 

First,  it  is  in  accordance  with  the  genius  of  aiithmetic;  we 
explain  an  arithmetical  subject  upon  arithmetical  principles. 
By  the  synthetic  method  we  leave  the  subject  of  arithmetic, 
and  bring  in  geometry  to  explain  arithmetic.  Should  it  be 
said  in  reply  that  by  the  analytic  method  we  are  explaining 
arithmetic  by  algebra,  let  it  be  remembered  that  algebra  has 
been  called  "universal  arithmetic,"  and  that  all  the  algebra 
that  is  here  used  is  purely  arithmetical.  In  other  words, 
though  we  may  indicate  the  analysis  of  the  number  by  letters, 
the  idea  is  purely  an  arithmetical  one,  and  is  in  no  way  depend- 
ent upon  the  principles  of  algebra  as  different  from  arithmetic. 

Second,  I  hold  that  a  full,  complete,  and  thorough  insight 
into  the  subject  can  be  obtained  only  by  the  analytic  method. 
The  geometric  method  indicates  the  process,  as  well  as  the 
analytic;  but  the  analytic  method  shows  the  nature  of  the  pro- 
cess, it  exhibits  the  law  of  the  formation  of  the  square  or  cube 
as  a  pure  process  of  arithmetic ;  and  this  gives  a  deeper  in- 
sight into  the  subject  than  can  be  obtained  by  the  other  method. 
One  who  knows  evolution  only  by  the  synthetic  method,  does 
not  know  it  thoroughly. 

Third,  the  analytic  method  is  general;  it  will  explain  the 
method  of  extracting  all  roots.  The  geometrical  method  is 
special;  it  enables  us  to  extract  the  square  and  cube  roots 
only.     Thus,  the  square   root   is  regarded  as   the   side   of  a 


278 


THE    PHILOSOPHY    OF    ARITHMETIC. 


Fig.  1. 


square,  the  cube  root  as  the  side  of  a  cube;  but  we  have  no 
geometrical  conception  of  the  fourth  root,  no  figure  correspond- 
ing to  the  fourth  power,  and  therefore  no  idea  of  a  fourth  root; 
and  so  on  for  the  higher  roots. 

In  respect  of  the  comparative  difficulty  of  the  two  methods, 
it  may  be  remarked  that  it  is  generally  supposed  that  the  syn- 
thetic method  is  much  easier  than  the  analytic.  This,  however, 
I  very  much  doubt;  and  this  opinion  is  founded,  not  only  upon 
theory,  but  also  upon  the  experience  of  those  who  have  tried 
both  methods.  I  believe  that  a  thorough  knowledge  of  the 
subject  can  be  gained  much  sooner  by  the  analytic  than  by  the 
synthetic  method.  My  observation  has  been  that  pupils  often 
are  able  to  run  over  the  geometrical  explanation  without  really 
understanding  it.  It  is,  therefore,  recommended  that  the  ana- 
lytic method  be  introduced  into  our  text-books  and  systems 
of  instruction. 

The  so-called  synthetic  methods  of 
evolution  may  also  be  presented  in 
an  analytic  form.  Thus,  instead  of 
adding  to  the  square  A,  page  271, 
we  can  begin  with  the  large  square, 
take  out  the  square  A,  then  obtain 
the  width  of  the  rectangles  and  the 
dimensions  of  the  corner  square,  and 
then  subtract.  Indeed,  this  seems 
the  more  natural  method,  and  is  now 
being  adopted  by  American  writers. 
When  thus  presented,  it  would  be 
better  to  call  the  two  methods  the  al- 
gebraic and  geometric  methods. 

The  same  may  be  illustrated  in  the 
extraction  of  the  cube  root.  Let  Fi^ 
1  represent  a  cube  which  contains 
91125  cubic  units.  Taking  out  the 
cube,  A  (403=  64000),  we  have  a  solid,  Fig.  2,  representing 
27125  cubic  units.     This  solid  consists  principally  of  the  three 


EVOLUXION. 


279 


slabs,  B,  C,  and  D,  each   40  units  in  Fig  3. 

length  and  breadth.     Dividing  27125 

by  the  sum  of  the  areas  of  a  face  of 

each,  (3X402  =  4800),  we  find  their 

thickness    is   5  units.     Removing   the 

slabs,   there  remain  three  solids,  Fig. 

3,  each  40  units  by  5  units,  hence  the 

surface  of  a  face  of  the  three  is  3  X  40 

X  5  =  600  square  miles. 

Removing  E,  F,  and  G,  there  re- 
mains the  small  cube  H,  Fig.  4,  the 
surface  of  one  of  whose  faces  is  52  = 
25  square  units.  Multiplying  the  sum 
of  all  these  surfaces  by  the  common 
thickness,  5,  we  have  (4800  +600+25) 
X  5  =  27125  cubic  units. 

New  Method  of  Cube  Root. — I  will  now  present  a  method 
of  extracting  cube  root  which  is  much  more  convenient  than  the 
ordinary  one.  The  simplification  consists  in  finding  a  general 
method  of  obtaining  the  trial  and  true  divisors,  so  that  any  one 
divisor  may  be  used  in  obtaining  the  next  following  divisor.  In 
the  operations  the  trial  divisor  is  indicated  by  t.  d.,  and  the  true 
divisor  by  T.  D.,  the  local  value  of  the  terms  being  distinguished 
by  their  position.  The  reason  for  the  method  of  obtaining  the 
trial  and  true  divisors  may  be  readily  shown  by  the  formula. 

1.  Extract  the  cube  root  of  14706125. 

Solution We  find  as  before  the  number  of  terms  in  the  root 

and  the  first  term  of  the  root  and  cube,  subtract  and  bring  down 
the  first  period. 

We  then  find  as  before  the  trial  divisor,  12,  by  taking  3  times 
the  square  of  the  first  term,  and,  dividing,  find  the  second  term 
of  the  root  to  be  4.  We  then  take  3  times  the  product  of  the  first 
and  second  terms  and  the  square  of  the  second  term,  and  add 
these  to  the  trial  divisor  as  a  correction  to  obtain  the  true  divisor ; 
1456.  We  then  multiply  1456  by  4,  and  subtract  and  bring 
down  the  next  period. 


280 


THE   PHILOSOPHY   OF   ARITHMETIC. 


OPERATION. 

14-706-125(245 

8 

12 t.  d. 

6706 

24   L 

16-1 

1456 
16 

T.  D. 

5824 

1728 t.  d. 

882125 

360 

25- 

176425  t.  d. 

882125 

To  find  the  next  trial  divisor, 
we  take  the  square  of  the  last  term, 
which  is  16,  and  add  it  to  the 
previous  true  divisor  and  the  two 
corrections  (which  were  added  to 
the  previous  trial  divisor),  and  we 
have  1728  as  the  next  trial  divisor. 

To  find  the  true  divisor,  we  add 
3  times  the  product  of  the  last 
term  of  the  root  into  the  previous 
part  of  the  root,  and  also  the  square 
of  the  last  term,  and  have  176425 
for  the  true  divisor.     Multiplying  by  5,  we  have  882125. 

The  method  is  indicated  in  the  following  formulas,  which  show 
the  formation  of  the  trial  and  true  divisors. 

1.  True  Divisor  =  Trial  Divisor  +  Product  +  Square. 

2.  Trial  Divisor=  Square+True  Divisor+Corrections. 
To  show  the  method  with  large  numbers,  extract  the  cube  root 

of  145780276447. 

Solution — We  find  the  first 
term  of  the  root,  the  first  trial 
divisor,  and  the  first  true  di- 
visor, as  before. 

To  find  the  second  trial  di- 
visor, we  take  the  sum  of  the 
square  of  2,  the  true  divisor, 
and  the  previous  correction, 
and  we  have  8112.  We  find  the 
next  true  divisor  by  adding  the 
usual  corrections  to  the  trial 
divisor,  and  have  820596. 

We  find  the  third  trial  di- 
visor by  taking  the  sum  of  the 
square  of  6,  the  previous  true 
divisor,  and  the  corrections,  and  have  830028.  We  find  the  next 
true  divisor  as  before. 


OPERATION. 

145-780-276-447(5263 
125 

75 

30 
4_ 

t. 

d. 

20780 

7804 
4 

T.D. 

15608 

8112 

936 
36- 

t. 

d.  i 

>172726 

820596 
36 

T.D. 

1923576 

830028 

4734 
9- 

t.  d. 

249150447 

83050149  t.  d. 

249150447 

EVOLUTION. 


281 


We  present  another  method  involving  the  principle  of  using  the 
previous  work  for  obtaining  trial  and  complete  divisors.  A  part 
of  this  method  is  easily  remembered  by  the  formulas. 

Complete  Divisor  =  Trial  Divisor  +  Correction. 

Trial  Divisor  =  Correction  -f-  Com.  Divisor  +Square. 

The  method  is  indicated  in  finding  the  cube  root  of  14706125. 

Solution. — We  find  the  number  of  figures  in  the  root,  and  the 
first  term  of  the  root,  as  in  the  preceding  methods. 

We  write  2,  the  first  term 

1st  Col. 

2 

4 

6T 

8 
725~ 


operation. 


2d  Col. 
12.  .  t.  d. 

-256 


14-706-125(245 

8 


1456  C  D. 

16 
1728..  t.d. 

3625 


6706 

5824 


176425  c.d. 


882125 


882125 


of  the  root,  at  the  left  at  the 
head  of  Col.  1st ;  3  times  its 
square  with  two  dots  an- 
nexed, at  the  head  of  Col. 
2d  ;  its  cube  under  the  left- 
hand  period ;  then  subtract 
and  annex  the  next  period 
for  a  dividend,  and  divide 
it  by  the  number  in  Col.  2d, 
as  a  trial  divisor,  for  the  second  term  of  the  root. 

We  then  take  2  times  2,  the  first  term,  and  write  the  product, 
4,  in  Col.  lst3  under  the  2,  and  add  ;  then  annex  the  second 
term  of  the  root  to  the  6  in  Col.  1st,  making  64,  and  multiply  64 
by  4  for  a  correction,  which  we  write  under  the  trial  divisor  ;  and 
adding  the  correction  to  the  trial  divisor,  we  have  the  complete 
divisor,  1456.  We  then  multiply  1456  by  4,  subtract  the  product 
from  6706,  and  annex  the  next  period  for  a  new  dividend. 

We  then  square  4,  the  second  figure  of  the  root,  write  the 
square  under  the  complete  divisor ;  and  add  the  correction,  the  com- 
plete divisor  and  the  square  for  the  next  trial  divisor,  which  we 
find  to  be  1728.  Dividing  by  the  trial  divisor  we  find  the  next 
term  of  the  root  to  be  5. 

We  then  take  2  times  4,  the  second  term,  write  the  product  8 
under  the  64,  add  it  to  64,  and  annex  the  third  term  of  the  root 
to  the  sum,  72,  making  725,  etc. 

A  part  of  this  method  can   be  easily  remembered    by  means 


282 


THE    PHILOSOPHY    OF    ARITHMETIC. 


of  the  following   formulas,  which  show  the  formation  of  the 
trial  and  complete  divisors  : 

1.  Trial  Divisor+Correction=Complete  Divisor. 

2.  Correction  4- Complete  Di  visor +Square=  Trial  Divisor. 
To  show  the  application  of  the  method  we  will  extract  the 

cube  root  of  41673648563. 


it  Col. 
3 

2d  Col. 

27  . .  t.  d. 

41-673-648'563(3467 

91*7 

6 

r 3'16              r" 

14673 

~94 

3076~c.  D. 

8 

16 

12304 

1026 

3468  . .  t.  d. 

2369648 

12 

r  6156 

10387 

352956  C.  D. 

36 

2117730 

359148  ..t.d. 

251912563 

72709 

35987509c.  D. 

251912563 

Horner's  Method.— Horner's  Method  of  extracting  the 
cube  root  was  derived  from  a  method  of  solving  cubic  equa- 
tions invented  by  Mr.  Horner,  of  Bath,  England.  It  was  first 
published  in  the  Philosophical  Transactions  for  1819;  under 
the  title  of  "A  New  Method  of  Solving  Numerical  Equations 
of  all  orders  by  Continuous  Approximations."  Its  inventor, 
Mr.  W.  G.  Horner,  was  a  teacher  of  mathematics  at  Bath ;  he 
died  in  1837.  It  is  considered  one  of  the  most  remarkable 
additions  made  to  arithmetic  in  modern  times.  DeMorgan 
says  that  the  first  elementary  writer  who  saw  the  value  of 
Horner's  method  was  J.  R.  Young,  who  introduced  it  in 
an  elementary  treatise  on  algebra,  published  in  1826.  Among 
the  first  to  introduce  it  into  arithmetic  in  this  country  was 
Prof.  Perkins,  of  New  York. 

This  method  differs  from  both  of  those  already  explained, 
and  possesses  merits  which  strongly  recommend  it  for  general 


EVOLUTION.  283 

adoption.  It  is  very  concise — the  root  of  a  large  number  can 
be  extracted  with  one-half  of  the  work  required  by  the  old 
method.  Its  conciseness  arises  from  the  fact  that  it  proceeds 
upon  a  principle  which  enables  us  to  make  use  of  work  already 
obtained,  while  the  old  method  requires  new  calculations  every 
time  we  find  a  trial  or  true  divisor.  In  other  words,  it  is  an 
organized  method  by  which  the  work  is  so  economized  that  no 
operations  are  superfluous,  but  each  result  obtained  is  made 
use  of  in  obtaining  a  subsequent  result. 

It  is  entirely  general  in  its  character,  applying  to  the  extrac- 
tion of  all  the  higher  roots.  This  method  can  be  explained 
both  aDalytically  and  synthetically.  It  is  presented  in  several 
of  the  higher  arithmetics,  and  need  not  be  stated  here.  It  is 
more  difficult  to  remember  than  either  of  the  other  methods, 
and  this  is  perhaps  the  principal  objection  to  its  general  adop- 
tion. The  "  New  Methods  "  for  cube  root — they  do  not  apply  to 
higher  roots — are,  however,  preferred  to  Horner's  Method,  being 
quite  as  concise,  and  much  more  readily  acquired  and  remem- 
bered. 

Approximate  Roots. — The  invention  of  rules  for  approxi- 
mating to  the  square  and  other  roots  of  numbers,  where  those 
roots  are  surds,  was  a  favorite  speculation  with  earlier  writers 
on  arithmetic  and  algebra.  These  rules  will  be  most  readily 
understood  and  their  relative  values  seen  by  stating  them  in 
algebraic  language. 

1.  The  rule  given  by  the  Arabs  is  expressed  by  the  formula, 

x 

x/(a2+x)=a+  — 

2lCL 

This  approximation  gives  the  root  in  excess ;  but  to  increase 
its  accuracy,  we  may  repeat  the  process,  making  use  of  the 
root  obtained.  This  is  the  rule  given  by  Lucas  di  Borgo,  and 
subsequently  by  Tartaglia,  who  derived  it  in  common  with  the 
rest  of  his  countrymen  from  Leonard  of  Pisa. 


284  THE   PHILOSOPHY   OF   ARITHMETIC. 

2.  The  rule  given  by  Juan  de  Ortega,  1534,  is  expressed  by 
the  following  formula : 

This  approximation  is  in  defect,  but,  generally  speaking,  more 
accurate  than  the  former. 

3.  The  third  method  of  approximation  was  proposed  by 
Orontius  Fineus,  Professor  of  mathematics  in  the  university 
of  Paris,  and  who  long  enjoyed  an  uncommon  reputation  in 
consequence  of  his  having  introduced  the  knowledge  of  the 
mathematics  of  Italy  among  his  countrymen.  His  method 
consisted  in  adding  2,  4,  6,  or  any  even  number  of  ciphers  to  the 
number  whose  root  was  required,  and  then  reducing  the  num- 
ber expressed  by  the  additional  figures  of  the  root  resulting 
from  these  ciphers,  to  sexagesimal  parts  of  an  integer.  Thus, 
in  extracting  the  square  root  of  10,  he  would  get  3  1 162,  which 
reduced  to  sexagesimals,  became  3.  9'.  43".  12//r. 

This  is  the  most  remarkable  approximation  to  the  invention 
of  decimals  which  preceded  the  age  of  Stevinus.  If  the 
author  had  stopped  short  at  the  first  separation  of  the  digits 
in  the  root,  it  would  have  expressed  the  square  root  of  10  to 
3  decimal  places;  but  the  influence  of  the  use  of  sexagesimals 
diverted  him  from  this  very  natural  extension  of  the  decimal 
notation,  and  retarded  for  more  than  half  a  century  this  im- 
provement in  the  science  of  numbers. 

The  method  of  Fineus  excited  the  attention  of  contempora- 
neous mathematicians,  who  in  adopting  it,  however,  did  not 
reduce  the  result  to  sexagesimals,  but  merely  subscribed,  as  a 
denominator  to  the  whole  not  considered  as  integral,  1  with  half 
as  many  ciphers  as  had  been  added  in  the  operation,  giving 
\/10=f  ooo-  it  is  under  this  form  that  it  is  noticed  by  Tar- 
taglia  and  Recorde.  Pelletier  also,  a  pupil  of  Orontius  Fineus, 
after  noticing  the  second  of  the  two  methods  of  approxima- 
tion, describes  this  as  more  accurate  and  less  tedious  than  any 
other. 

Methods  of  approximation  were  also  quite  numerous  for  the 


EVOLUTION.  285 

extraction  of  the  cube  root.  That  of  Lucas  di  Borgo  may  be 
seen  from  the  formula, 

which  Tartaglia  says  he  got  from  Leonard  of  Pisa,  who  had  it 
from  the  Arabians;  and  he  expresses  his  surprise  that  he 
should  have  committed  so  grievous  an  error,  unless  he  had  done 
so  without  consideration. 

The  method  of  Orontius  Fineus  is  represented  by  the  follow- 
ing formula : 

x 
^/(aHx)=a+  — - 

which  errs  as  much  in  excess  as  that  of  Di  Borgo  in  defect. 
The  method  of  Cardan  is  indicated  by  the  formula, 

which  Tartaglia  criticises  with  great  bitterness,  as  might  nat- 
urally be  expected  from  one  who  had  been  so  treacherously 
defrauded  by  him  of  an  important  discovery,  the  general 
method  of  solving  cubic  equations.  His  own  method  is  rep- 
resented by  the  formula, 

which,  though  more  accurate  than  that  of  Cardan,  errs  in  defect 
while  the  other  erred  in  excess. 

In  later  times,  methods  of  approximation  have  been  proposed 
which  give  results  much  more  accurate  than  any  of  the  pre- 
ceding. One  of  the  very  best  that  we  have  met  is  the  follow- 
ing, given  by  Alexander  Evans,  in  the  January  number  of  The 
Analyst,  18*76 : 

For  square  root,—  -f- 

N   .    2r 
For  cube  root,  3^2  +  ~jT 

N        n-1 

For  nth  root,  — —. .  4- r 

nrn~l  n 


286  THE     PHILOSOPHY    OF    ARITHMETIC. 

To  illustrate  these  formulas  we  will  extract  the  square  root 
of  2  and  the  cube  root  of  6.  Suppose  the  square  root  of  2  is 
nearly  1.4,  then  r=1.4,  and  substituting  in  the  formula  we 
have 

iV      r         2  1    14        99 

^  +  2~2(Hy+2'l0"==Y0  =  1-4142+ 
which  is  the  correct  root  to  four  places ;  and  by  substituting  f| 
in  the  formula  we  get  the  root  correct  to  eight  places. 

In  extracting  the  cube  root  of  6,  suppose  that  r=1.8,  then 
substituting  in  the  formula  we  have 

3r         3  3(i|)2  +      3     _81+5  T 

which  is  true  to  three  decimal  places.  The  method  cannot  be 
relied  upon,  however,  to  give  many  correct  terms  in  the  ap- 
proximation. In  applying  it  to  the  cube  root  of  3,  regarding 
1.4  as  the  value  of  r,  we  obtain  for  the  root,  1.44353,  which 
is  true  to  only  two  places.  If  we  then  take  1.44  as  the  value 
of  r,  we  shall  find  the  next  approximation  to  be  1.442253, 
which  is  true  to  four  places.  If  we  take  r=1.5  as  the  cube  root 
of  4,  the  formula  gives  the  first  approximation  1.5925,  which  is 
true  to  only  the  first  decimal  place.  If  we  had  taken  r=1.6, 
we  would  have  obtained  1.5875,  which  is  correct  to  three 
places.  The  best  method  is  therefore  the  general  one;  for  a 
person  who  is  familiar  with  the  method  which  I  have  given 
under  the  name  of  the  New  Method  will  extract  the  root  more 
rapidly  than  he  can  with  the  approximate  methods,  and  may 
be  always  certain  of  the  correctness  of  his  result. 


Part  III. 
COMPARISON. 


SECTION  I. 
RATIO  AND  PROPORTION. 

SECTION  II. 
THE  PROGRESSIONS. 

SECTION  III. 
PERCENTAGE. 

SECTION  IV. 
THEORY  OF  NUMBERS. 


SECTION    I. 

RATIO  AND  PROPORTION. 


19 


I.     Introduction. 


II.     Nature  of  Ratio. 


III.     Nature  op  Proportion 


IV.     Application  of  Proportion. 


Y.     Compound  Proportion,  etc. 


VI.     History  of  Proportion. 


CHAPTER  I. 
INTRODUCTION   TO   COMPARISON. 

ARITHMETIC  consists  fundamentally  of  three  processes ; 
Synthesis,  Analysis,  and  Comjjarison.  Synthesis  and 
Analysis  are  mechanical  processes  of  uniting  and  separating 
numbers  ;  Comparison  is  the  thought  process  which  directs  the 
general  processes  of  synthesis  and  analysis,  and  unfolds  the 
various  particular  processes  contained  in  them.  Comparison 
also  gives  rise  to  several  processes  which  do  not  grow  out  of 
the  general  operations  of  synthesis  and  analysis,  but  which 
have  their  origin  in  the  thought  process  itself  The  principal 
processes  originating  in  Comparison,  are  Ratio,  Proportion, 
Progression,  Percentage,  Reduction,  and  the  Properties  of 
Numbers.  The  particular  manner  in  which  these  processes 
originate  will  appear  from  the  following  considerations. 

If  two  numbers  be  compared  with  each  other,  we  perceive 
a  definite  relation  existing  between  them,  and  the  measure  of 
this  relation  is  called  Ratio.  Numbers  may  be  compared  in 
two  ways:  first,  by  inquiring  how  much  one  number  is  greater 
or  less  than  another ;  and  secondly,  by  inquiring  how  many  times 
one  number  equals  another.  Thus,  in  comparing  6  with  2,  we 
see  that  6  is  four  more  than  2,  and  also  that  6  is  three  times  % 
These  relations,  expressed  numerically,  give  us  the  ratio  of  the 
numbers.  The  former  is  called  arithmetical  ratio;  the  latter, 
geometrical  ratio.  The  term  ratio  is  generally  restricted,  how- 
ever, to  a  geometrical  ratio,  and  it  will  be  thus  used  here. 

The  comparison  of  ratios  gives  rise  to  several  distinct  pro- 
cesses called  Proportion.     If  two  equal  ratios  be  compared, 

(291) 


292  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  numbers  producing  the  ratios  being  retained  in  the  com- 
parison, we  have  what  we  call  a  Geometrical  Proportion,  or 
simply  a  Proportion.  When  the  ratios  are  simple,  we  have  a 
Simple  Proportion ;  when  one  or  both  of  the  ratios  are  com- 
pound, we  have  a  Compound  Proportion. 

If  we  wish  to  divide  a  number  into  several  equal  parts,  bear- 
ing a  certain  relation  to  each  other,  we  have  a  process  called 
Partitive  Proportion.  If  we  wish  to  combine  numbers  in 
certain  definite  relations,  we  have  a  process  called  Medial  Pro- 
portion, usually  known  as  Alligation.  If  we  compare  num- 
bers so  that  each  consequent  is  of  the  same  kind  as  the  next  an- 
tecedent, we  have  a  process  known  as  Conjoined  Proportion. 

If  we  have  a  series  of  numbers  differing  by  a  common  ratio, 
we  may  investigate  such  a  series  and  ascertain  its  laws  and 
principles;  thus  arises  the  subject  of  Progressions.  If  the 
ratio  is  arithmetical,  the  progression  is  called  an  Arithmetical 
Progression;  if  the  ratio  is  geometrical,  the  progression  is 
called  a  Geometrical  Progression. 

Again,  as  was  shown  in  the  Logical  Outline  of  arithmetic,  we 
may  take  some  number  as  a  basis  of  comparison,  and  develop 
the  relations  of  numbers  with  respect  to  this  basis.  It  has 
been  found  convenient  in  business  transactions  to  use  one  hun- 
dred as  such  a  basis  of  comparison,  which  gives  rise  to  the 
subject  of  Percentage.  In  Fractions  and  Denominate  Numbers 
we  have  units  of  different  values  under  the  same  general  kind 
of  quantity.  By  comparing  these,  it  is  seen  that  we  may 
pass  from  a  unit  of  one  value  to  one  of  a  greater  or  less  value, 
and  thus  arises  the  process  of  Reduction.  When  we  pass  from 
a  less  to  a  greater  unit  the  process  is  called  Reduction  Ascend- 
ing ;  when  we  pass  from  a  greater  to  a  less  unit,  the  process 
is  called  Reduction  Descending. 

By  a  comparison  of  numbers,  we  may  also  discover  certain 
properties  and  principles  which  belong  to  numbers  per  se,  and 
also  other  properties  and  principles  which  have  their  origin  in 
the  Arabic  system  of  notation.     Such  principles  may  be  em- 


INTRODUCTION    TO    COMPARISON.  293 

braced  under  the  general  head  of  the  Properties  of  Numbers. 
It  is  thus  seen  that  several  divisions  of  the  science  of  numbers 
are  not  contained  in  the  original  processes  of  synthesis  and 
analysis — that  is,  of  addition  and  subtraction — but  have  their 
roots  in  and  grow  out  of  the  thought-process  of  comparison. 
These  several  subjects,  evolved  from  the  comparison  of  num- 
bers, will  be  considered  in  the  order  in  which  they  hare  beeL> 
mentioned. 


CHARTER  II. 
NATURE   OP  RATIO. 

KATIO  originates  in  the  comparison  of  numbers.  It  is  the 
numerical  measure  of  their  relation.  From  it  arise  some 
of  the  most  important  parts  of  arithmetic,  as  proportion,  pro- 
gressions, etc.  Its  importance,  and  the  inadequate  and  diverse 
views  held  concerning  it,  make  it  necessary  to  give  quite  a  care- 
ful and  thorough  discussion  of  the  subject. 

Definition. — Ratio  is  the  measure  of  the  relation  of  two 
similar  quantities.  This  definition  differs  in  one  respect  essen- 
tially from  that  usually  given.  Ratio  is  generally  defined  as 
"the  relation  of  two  quantities" — relation  and  ratio  being 
made  equivalent.  This  is  not  accurate,  or,  at  least,  not  suffi- 
ciently definite.  The  word  ratio  is  a  more  precise  term  than 
relation,  as  will  appear  from  the  following  illustration.  If  we 
inquire  what  is  the  relation  of  8  to  2,  the  natural  reply  is  "  8 
is  four  times  2  ;"  but  if  we  inquire  what  is  the  ratio  of  8  to  2, 
the  correct  reply  is  "four."  Ilere  the  ratio  four  is  the  num- 
ber which  measures  the  relation  of  8  compared  with  2.  It  is 
thus  seen  that  ratio  is  not  merely  the  relation  of  two  similar 
quantities,  but  the  measure  of  this  relation.  This  definition, 
presented  in  the  author's  own  text-books,  has  already  been  in- 
troduced by  one  or  two  writers,  and  seems  not  unworthy  of 
general  adoption. 

The  Terms. — A  ratio  arises  from  the  comparison  of  two 
similar  quantities.  These  quantities  are  called  the  terms  of  the 
ratio.  The  first  term  is  called  the  Antecedent ;  the  second  term 
is  called  the  Consequent.     The  antecedent  is  compared  with 

(294) 


NATURE   OF   RATIO.  295 

the  consequent;  the  consequent  is  the  basis  or  standard  of 
comparison.  Thus,  a  ratio  indicates  the  value  of  the  Orst 
quantity  as  compared  with  the  second  as  a  standard.  The 
ratio,  therefore,  expresses  how  many  times  the  consequent  must 
be  taken  to  produce  the  antecedent,  or  what  part  the  antece- 
dent is  of  the  consequent.  In  other  words,  it  answers  the 
question — the  antecedent  is  how  many  times  the  consequent,  or, 
the  antecedent  is  what  part  of  the  consequent?  From  this  it 
also  appears  that  the  ratio  equals  the  antecedent  divided  by 
the  consequent.  Thus,  the  ratio  of  6  to  3  is  2,  and  the  ratio  of 
3  to  6  is  J. 

Method  of  Ratio. — The  question  has  recently  been  raised 
whether  the  correct  method  of  determining  a  ratio  is  to  divide 
the  antecedent  by  the  consequent  or  the  consequent  by  the  an- 
tecedent. An  eminent  author  advocates  the  division  of  the 
consequent  by  the  antecedent,  and  this  method  has  been  adopted 
by  several  American  mathematicians.  The  old  method  some 
of  them  call  the  "  English  Method ;"  the  new  method,  the 
" French  Method."  The  so-called  "French  Method"  we  be- 
lieve to  be  incorrect  in  principle  and  inconvenient  in  practice. 
The  correct  method  of  finding  the  ratio  of  two  numbers  is  to 
divide  the  antecedent  by  the  consequent.  Several  reasons  will 
be  given  in  favor  of  the  correctness  of  this  method,  which  seem 
to  us  conclusive.  For  convenience  in  the  discussion,  let  us 
distinguish  the  two  methods  as  the  Old  and  the  New  method. 

1.  Nature  of  Ratio. — First,  I  think  the  correctness  of  the 
Old  Method  will  appear  from  the  nature  of  ratio  itself.  If  we 
inquire  "  What  is  the  relation  of  8  to  2  ?"  the  natural  reply  is, 
"  8  is  four  times  2."  Here  the  number  four  is  the  measure  of 
the  relation ;  hence  the  ratio  of  8  to  2  is  four.  If  the  inquiry 
is,  "What  is  the  relation  of  2  to  8?"  the  natural  reply  is  "  2  is 
one-fourth  of  8 ;"  hence  in  this  case,  the  ratio  is  one-fourth. 
From  this  view  of  the  subject  it  follows  that  the  correct  method 
cf  determining  a  ratio  is  to  divide  the  antecedent  by  the  conse- 
quent, and  not  the  consequent  by  the  antecedent. 


296  THE   PHILOSOPHY    OF   ARITHMETIC. 

If  I  ask  the  relation  of  8  to  2,  it  would  be  illogical  to  reply,  "  a 
is  one-fourth  of  8,"  for  this  does  not  answer  my  question.  In 
giving  the  reply,  that  number  should  be  used  first  in  making  the 
comparison  which  was  used  first  in  the  question,  and  it  would 
be  illogical  and  absurd  to  invert  the  order ;  yet  this  is  really  what 
those  who  advocate  the  other  method  must  do.  If  the  ratio  of 
8  to  4  is  one-half,  then  when  1  ask  the  question,  "What  is  the 
relation  of  8  to  4  ?"  they  must  say,  "4  is  one-half  of  8,"  unless 
it  be  supposed  that  they  would  say,  "  8  is  one-half  of  4." 

This  may  be  impressed  by  an  illustration  suggested  by  Prof. 
Dodd.  Of  two  persons,  A  and  B,  suppose  A  to  be  the  father 
and  B  the  son.  Now  if  the  question  be  asked,  "  What  is  the 
relation  of  A  to  B  V  the  correct  reply  is  "A  is  the  father  of 
B,"  and  it  would  be  inconsistent  to  answer,  UB  is  the  son  of 
A,11  for  that  is  the  reply  to  the  question,  "  What  is  the  relation 
of  B  to  A  ?"  The  same  holds  in  regard  to  the  comparison  of 
numbers,  and  with  even  greater  force,  since  it  is  necessary  to 
be  more  explicit  in  science  than  in  ordinary  conversation. 
Hence,  if  the  question  is  asked,  "What  is  the  relation  of  8  to 
2  ?"  the  correct  reply  is,  "  8  is  four  times  2  ;"  from  which  we 
see  that  the  ratio  is  four.  It  is  clear,  then,  that  the  ratio  of 
two  numbers,  which  is  the  measure  of  the  relation  of  the  first 
to  the  second,  is  equal  to  the  first  divided  by  the  second. 

2.  Law  of  Comparison. — The  true  method  of  determining 
a  ratio  may  also  be  shown  by  the  nature  and  object  of  the  com- 
parison. The  law  of  comparison  is  to  compare  the  unknown 
with  the  known ;  thus,  we  logically  write  #=4,  and  not  4=x. 
Now,  in  a  ratio,  one  number  is  made  the  basis  of  comparison, 
the  object  being  to  comprehend  or  measure  the  other  Dumber 
by  its  relation  to  the  basis.  In  this  sense  the  basis  may  be 
regarded  as  the  known  quantity,  and  the  other  number  as  the 
unknown  quantity.  Now  the  unit  is  the  basis  of  all  numbers; 
it  is  the  standard  by  which  all  numbers  are  measured ;  we  un- 
derstand a  number  only  as  we  know  its  relation  to  the  unit. 
When  any  number,  as  8,  is  presented  to  the  mind,  we  compare 


NATURE   OF   RATIO.  297 

it  with  the  unit,  not  the  unit  with  it.  The  inquiry  is,  8  is  how 
many  times  one?  hence  8  is  the  first  number  named  in  the  com- 
parison ;  it  is,  therefore,  the  antecedent,  and  the  ratio  is  the 
quotient  of  the  antecedent  by  the  consequent.  The  advocates 
of  the  new  method  of  ratio  would  have  us  compare  the  1  with 
the  8,  the  unit  of  measure  with  the  thing  to  be  measured,  the 
known  with  the  unknown.  This  is  not  only  awkward,  but  it 
is  directly  opposed  to  the  established  principles  of  logical 
thought. 

3.  Authority. — One  of  the  strongest  arguments  in  favor  of 
the  division  of  the  first  term  by  the  second  is  the  usage  of 
eminent  mathematicians.  That  signification  of  scientific  terms 
which  custom  has  fixed  should  not  be  changed  but  for  the 
strongest  reasons.  From  the  earliest  periods  of  science,  math- 
ematicians have  divided  the  antecedent  by  the  consequent.  It 
was  the  method  employed  by  Euclid,  Pythagoras,  and  Archi- 
medes, the  three  great  mathematicians  of  antiquity;  and  by 
Newton,  LaPlace,  and  LaGrange,  the  three  great  mathemati- 
cians of  modern  times.  The  English  and  German,  and  nearly 
all  the  French  mathematicians,  employ  this  method,  and  have 
done  so  from  the  earliest  periods.  One  or  two  French,  and  a 
few  American  authors  have  adopted  the  New  Method ;  but 
with  these  few  exceptions,  the  Old  Method  is  the  method  of 
mathematicians  at  all  times  and  in  every  country  where  the 
ratio  of  numbers  has  been  employed. 

But  not  only  is  the  authority  of  numbers  upon  this  side  of 
the  question,  but  also  the  greater  weight  of  the  authority  of 
eminence.  The  practice  of  all  of  the  great  mathematicians  of 
every  age  is  in  favor  of  the  Old  Method.  In  its  favor  we  may 
mention  the  illustrious  names  of  Euclid,  Pythagoras,  Archi- 
medes, and  to  these  add  the  not  less  illustrious  names  of  Dio- 
phantus,  Newton,  Leibnitz,  LaPlace,  LaGrange,  the  Bernoullis, 
Legendre,  Arago,  Bourdon,  Carnot,  Barrow,  Ilerschel,  Bow- 
ditch,  Pierce,  etc.;  names  which  shed  a  lustre  over  their  country 
and  age,  and  which  are  symbols  of  grand  achievements  in  the 
13* 


298  THE    PHILOSOPHY    OF    ARITHMETIC. 

science.  All  the  great  works,  the  masterpieces  which  stand  as 
monuments  of  the  loftiest  triumphs  of  genius,  are  upon  this  side  of 
the  question.  The  Principia  of  Newton,  the  Mecanique  Celeste 
of  LaPlace,  the  Mecanique  Analytique  of  LaG range,  the 
Theorie  des  Nombres  of  Legendre,  the  Analytical  Mechanics 
of  Pierce,  all  employ  the  Old  Method.  Such  universal  agree- 
ment among  great  mathematicians  should  be  regarded  as  a  final 
settlement  of  the  matter. 

4.  Inconvenience  of  the  Change. — Again,  the  Old  Method 
cannot  be  changed  without  confusion.  There  are  definitions 
in  science  which  involve  the  idea  of  ratio,  and  a  correct  appre- 
hension of  these  definitions  requires  a  precise  idea  of  ratio. 
These  definitions  are  founded  upon  the  Old  Method  of  ratio ; 
hence,  if  we  change  the  method  of  determining  ratio,  we  shall 
either  have  a  wrong  idea  of  the  subjects  defined,  or  else  the 
definitions  must  be  changed.  The  latter  would  be  almost  a 
practical  impossibility,  since  they  have  become  fixed  forms  in 
scientific  language.  Science  has  embalmed  certain  definitions, 
and  it  would  seem  almost  like  sacrilege  to  disturb  them. 

Among  these  definitions  may  be  mentioned  those  of  specific 
gravity,  differential  co-efficient,  index  of  refraction,  and  the 
geometrical  symbol  n.  The  specific  gravity  of  a  body  is  defined 
to  be  the  ratio  of  its  weight  to  the  weight  of  an  equal  volume 
of  some  other  body  assumed  as  a  standard.  The  index  of  re- 
fraction is  the  ratio  of  the  sine  of  the  angle  of  incidence  to  the 
sine  of  the  angle  of  refraction.  The  differential  co-efficient  is 
the  ratio  of  the  increment  of  the  function  to  that  of  the  varia- 
ble. The  geometrical  symbol  tz  is  the  ratio  of  the  circumfer- 
ence to  the  diameter.  These  definitions  have  the  authority  of 
the  great  masters,  and  will,  without  doubt,  remain  as  they  are. 
One  or  two  of  them  have  been  changed  by  the  advocates  of 
the  New  Method,  but  such  changes  will  hardly  extend  beyond 
their  own  text-books. 

5.  Origin  of  Symbol. — It  may  further  be  remarked  that  the 
assumed  origin  of  the  symbol  of  ratio  is  in  favor  of  the  method 


NATURE   OF   RATIO.  299 

here  advocated.  It  is  said  that  the  symbol  of  ratio  is  derived 
from  that  of  division ;  that  is,  that  :  is  the  symbol  -v-  with  the 
horizontal  line  omitted.  The  symbol  of  division  indicates  that 
the  quantity  before  it  is  to  be  divided  by  the  one  following  it; 
hence  if  the  theory  of  the  origin  is  true,  it  indicates  that  prima- 
rily the  ratio  of  two  numbers  was  the  quotient  of  the  first 
divided  by  the  second ;  and  this  primary  method  should  be 
followed,  unless  there  are  good  reasons  to  the  contrary. 

In  this  connection  I  remark  that  the  Old  Method  of  ratio 
gives  us  the  simplest  idea  of  a  proportion.  A  proportion  is 
an  equality  of  ratios,  and  this  idea  is  most  clearly  expressed 
thus:  6-h3=8-=-4.  With  the  other  method  of  ratio,  this  sim- 
ple idea  of  a  proportion  cannot  be  presented.  Whether  the 
symbol  :  is  a  modification  of  -h,  is,  I  presume,  not  definitely 
known.  It  is  so  asserted  by  some  authors ;  but  so  far  as  I  can 
learn,  it  is  not  known  as  a  historical  fact.  It  seems  very  reason- 
able, however,  and  in  some  old  German  works  I  have  noticed 
that  the  symbol  of  division  is  used  for  indicating  the  ratio  of 
numbers. 

The  "French  Method,"  inappropriately  so  called. — These 
two  methods  of  ratio  have  been  distinguished  by  the  names 
"English  Method,"  and  "French  Method;"  the  Old  Method 
being  called  the  "English  Method,"  and  the  New  Method  the 
"French  Method."  These  names  were  first  applied,  I  think, 
by  Prof.  Ray,  although  others  had  previously  stated  that  the 
French  mathematicians  made  use  of  the  one  and  the  English 
mathematicians  of  the  other  method.  Both  of  these  names  are 
founded  in  error.  The  "  French  Method  "  is  not  used  by  the 
French ;  the  general  custom  of  the  French  mathematicians 
is  opposed  to  it.  Lacroix  is  the  only  mathematician  of  any 
eminence  who,  so  far  as  I  have  examined,  employs  it.  The 
"  English  Method"  is  not  conGned  to  the  English,  but  it  is  used 
by  French,  Germans,  Prussians,  and  Austrians,  in  fact,  by  the 
mathematicians  of  all  countries,  and  is,  therefore,  incorrectly 
named  the  English  Method. 


300  THE   PHILOSOPHY   OF   ARITHMETIC. 

Nearly  all  the  mathematicians  of  France,  it  has  been  said, 
employ  the  so-called  English  Method,  and  all  of  the  most  emi- 
nent ones  do  so.  Among  these  may  be  mentioned  LaPlace, 
LaGrange,  Legendre,  Bourdon,  Vernier,  Comte,  Biot,  Carnot, 
Arago,  etc.  In  proof  of  this,  I  will  quote  from  some  of  their 
own  works.  M.  Bourdon,  in  his  Arithmetic,  page  222,  says, 
"Par  exemple,  le  rapport  de  24  a  6  est  ^,  ou  4 ;  et  celui  de  6 
a  24  est  -fa,  ou  \.  Legendre,  in  his  Geometry,  Book  IV.,  Prop. 
XIV.,  says,  "done  le  rapport  de  la  circonference  au  diametre 
designe  ci-dessus  par  it  =3.1415926."  Vernier,  in  his  Arith- 
metic, page  118,  says,  "  comme  la  raison  est  le  quotient  qu'  on 
obtient  quand  on  divise  V  antecedent  par  le  consequent."  Other 
authors  might  be  quoted,  but  these  are  sufficient  to  show  that 
the  so-called  French  Method  is  not  the  method  of  the  French. 
Legendre  and  Bourdon  are  especially  referred  to,  since  some 
popular  American  text-books,  supposed  to  be  translations  from 
these  authors,  employ  the  New  Method,  and  have  been  instru- 
mental in  leading  quite  a  large  number  of  American  authors 
and  teachers  to  adopt  that  method. 

In  turning  to  Lacroix,  we  see  a  departure  from  the  general 
usage  of  the  French  mathematicians.  In  his  Arithmetic,  which 
is  the  only  work  of  his  that  I  have  examined,  he  says,  page  85, 
in  comparing  the  numbers  13,  18,  130,  and  180,  we  see  '-que  le 
deuxieme  conlient  le  premier  autant  de  fois  que  le  quatrieme 
conlienl  le  troisieme;  et  Us  formenl  ainsi  ce  qu'on  appelle  une 
pro])orlion."  Notice  that  he  is  here  discussing  the  subject  of 
proportion,  and  not  the  subject  of  ratio  by  itself.  On  the  next 
page  he  remarks,  "Je  conlinuerai  de  prendre  le  consequent  du 
ra]>])ort  pour  le  numerateur  de  la  fraction  qui  exprime  le 
raj>pcrl  el  V  antecedent  pour  le  denominaleur." 

This  places  Lacroix  upon  the  opposite  side  of  this  question; 
and  it  is  clear  from  the  manner  in  which  he  expresses  himself, 
that  he  is  conscious  of  taking  a  position  not  authorized  by  the 
general  custom  of  his  countrymen.  I  think  it  can  readily  be 
seen  how  Lacroix  was  led  into  this  error      lie  commences  the 


NATURE    OF    RATIO.  301 

subject  with  a  problem  in  proportion,  which  he  solves  by  anal- 
ysis, and  then,  by  a  mistake  plausibly  drawn  from  the  process 
of  analysis,  seeming  to  think  that  the  analysis  dictates  a  divi- 
sion of  consequent  by  antecedent,  he  defines  his  terms  and  an- 
nounces his  method  of  ratio.  The  whole  discussion  is  as  illog- 
ical as  the  conclusion  is  incorrect.  He  begins  the  subject  with 
proportion  instead  of  ratio,  thus  inverting  the  whole  problem 
and  getting  the  method  of  ratio  inverted  also.  The  true  method 
is  to  begin  by  comparing  numbers,  determining  their  relations: 
and  then  comparing  their  relations,  make  a  proportion ;  the  first 
will  give  the  true  idea  of  Ratio,  and  the  second  of  Proportion. 

Answer  to  Arguments  in  Favor  of  the  New  Method.— This  dis- 
cussion would  be  imperfect  without  an  attempt  to  answer  some 
of  the  arguments  which  have  been  presented  in  favor  of  the 
so-called  "French  Method."  An  eminent  author  and  educator, 
who  has  done  more  for  the  adoption  of  the  New  Method  than 
any  other  person  in  this  country,  gives  a  formal  defense  of  it; 
a  few  of  his  arguments  I  will  notice.  His  first  argument,  which 
is  founded  upon  the  nature  of  comparison,  has  already  been 
answered  in  the  previous  discussion.  He  says,  in  comparing 
numbers,  "  the  standard  should  be  the  first  number  named ;" 
hence,  to  comprehend  8,  he  would  compare  the  basis  of  num- 
bers, or  1,  with  8,  instead  of  comparing  the  8  with  1,  that  is, 
the  number  with  the  basis.  The  mistake  he  makes  is  in  com- 
paring the  standard  with  the  thing  measured  ;  that  is,  the 
known  with  the  unknown ;  the  true  law  of  comparison  being  just 
the  reverse  of  this. 

This  will  be  readily  seen  in  continuous  quantity  which  can 
be  clearly  understood  only  by  comparing  it  with  some  definite 
part  of  itself  assumed  as  a  unit.  Thus,  suppose  a  period  of  time 
is  considered ;  it  is  clear  that  we  can  get  a  definite  idea  of  it  by 
comparing  it  with  some  fixed  unit,  as  a  day,  or  a  week,  or  a 
year.  In  these  cases  it  will  be  seen  that  we  do  not  compare  the 
unit  with  the  given  quantity,  as  the  author  quoted  would  main- 
tain, but  the  quantity  to  be  measured  with  the  unit  of  measure. 


802  THE    PHILOSOPHY    OF    ARITHMETIC. 

His  second  argument  is  that  the  New  Method  gives  a  con- 
venient rule  for  Proportion ;  the  fourth  term  being  equal  to  the 
third  term  multiplied  by  the  ratio  of  the  first  to  the  second. 
The  reply  is  that  the  Old  Method  gives  just  as  convenient  a  rule, 
namely,  "  The  fourth  term  equals  the  third  divided  by  the  ratio 
of  the  first  to  the  second."  His  third  argument  is,  that  in  a 
geometrical  progression  the  ratio  is  the  quotient  of  any  term 
divided  into  the  following  term.  This  is  the  most  plausible 
argument  advanced,  and  demands  special  notice.  If  it  be  true 
that  the  ratio  of  any  term  to  the  following  term  is  the  quotient 
of  the  second  divided  by  the  first,  then  it  is  true  that  we  here 
depart  from  the  general  method  of  ratio ;  but  still  it  would  not 
follow  that  the  general  method  of  ratio  should  be  changed  to 
harmonize  with  this  exceptional  case.  A  more  sensible  conclu- 
sion would  be  that  the  method  here  used  should  be  changed  to 
correspond  with  the  general  method.  That  the  general  should 
control  the  special  and  not  the  special  the  general,  is  a  fixed 
law  of  science.  Let  us  see,  however,  if  the  form  of  writing  a 
geometrical  progression  does  present  an  exception  to  the 
general  method  of  expressing  a  ratio. 

In  a  geometrical  progression,  the  ratio  is  the  measure  of  the 
relation  that  any  term  bears  to  the  preceding  term.  In  the 
series  1,  2,  4,  8,  etc.,  we  do  not  compare  the  1  with  the  2,  the  2 
with  the  4,  etc.,  to  determine  the  ratio,  as  will  appear  from  the  fol- 
lowing considerations.  Suppose,  for  illustration,  that  we  wish 
to  find  any  term  of  the  series,  as  the  5th  term,  would  we  not 
reason  thus :  the  5th  term  must  bear  the  same  relation  to  the 
4th,  that  the  4th  does  to  the  3d ;  and  since  the  4th  is  twice  the 
3d,  the  5th  term  must  be  twice  the  4th,  or  16.  Here  we  follow 
the  law  of  comparison,  by  comparing  the  unknown  with  the 
known,  and  reversing  the  apparent  order,  name  the  8  first  and 
the  4  after  it.  Should  we  write  the  comparison  out  in  full,  we 
would  have  5th  :  8  : :  8  :  4.  If  this  is  true,  then,  in  a  geometri- 
cal scries,  we  do  not  compare  a  term  with  the  following  term, 
but  rather  with  the  term  preceding  it      The  ratio  of  the  series, 


NATURE   OF   RATIO.  803 

it  thus  appears,  is  the  ratio  of  any  term  to  the  preceding  term, 
and  not  to  the  term  following  it.  In  other  words,  we  compare 
backward,  instead  of  forward,  as  in  ordinary  ratio ;  and  really 
divide  the  antecedent  of  the  comparison  by  the  consequent  tc 
obtain  the  ratio. 

Some  writers  explain  this  apparent  departure  from  the  gen- 
eral signification  of  ratio,  by  saying  that  in  a  geometrical  series 
we  express  the  "  inverse  ratio  of  the  terms."  Says  one,  "  It  is 
less  troublesome  to  express  the  common  ratio  inversely,  as  then 
one  number  will  suffice."  Says  another,  "Whenever  we  meet 
with  the  expression,  the  'ratio  of  a  geometrical  series,'  we  are 
to  understand  the  inverse  ratio."  It  seems  clearer  to  me  to 
say  that  the  order  of  writing  the  terms  is  in  opposition  to  the 
order  of  thought.  We  write  one  way  and  compare  another 
way.  If  the  expression  of  the  series  were  dictated  by  the 
idea  of  ratio,  we  would  write  it  from  the  right  toward  the  left. 

The  fact  is,  however,  that  in  a  geometrical  progression,  it  is 
the  rate  of  the  progression  that  we  consider,  rather  than  the 
ratio  of  the  terms ;  that  is,  the  rate  at  which  the  series  pro- 
gresses, and  this  term  would  be  preferable  to  ratio  in  this  con- 
nection. A  series  of  terms,  increasing  or  decreasing  by  a  common 
multiplier,  although  an  outgrowth  from  the  idea  of  ratio,  pre- 
sents an  idea  not  identical  with  that  of  ratio. 

This  distinction  is  actually  made  by  several  French  writers. 
They  use  the  different  words  rapport  and  raison ;  the  former 
to  express  the  ratio  of  two  numbers,  the  latter  to  denote  the 
rate  of  the  geometrical  series.  Thus  Bourdon,  in  his  Arith- 
metic, page  279,  says,  "  On  appelle  Progression  par  Quotient 
une  suite  de  nombres  tels  que  le  rapport  d'un  terme  quelconque 
a  celui  qui  le  precede  est  constant  dans  toute  Velendue  de  la  serie. 
Ce  rapport  constant,  qui  existe  entre  un  terme  el  celui  qui  le 
precede  immedialement  se  nomme  la  liaison  de  la  progression." 
Frof.  Henkle,  who  has  written  several  excellent  articles  upon 
this  subject,  quotes  Biot  to  the  same  effect.  He  says  of  a  geomet- 
rical progression,  "  Le  Rapport  de  chaque  terme  au  precedent  se 


304  THE    PHILOSOPHY    OF    ARITHMETIC. 

nomme  liaison."  It  will  thus  be  seen  that  some  of  the  French 
writers  distinguish  between  ratio  and  the  constant  multiplier  of 
a  progression,  and  should  the  word  rate  be  adopted  with  us, 
we  would  avoid  the  objection  of  this  seeming  departure  from 
the  general  signification  of  ratio. 

I  have  devoted  so  much  space  to  the  discussion  of  this  sub- 
ject, because  I  think  it  one  upon  which  there  should  be  uni- 
formity of  opinion  and  practice.  Several  of  our  most  popular 
elementary  text-books  on  mathematics  have  adopted  the  so- 
called  "French  Method,"  and  are  teaching  it  to  the  youth  of 
the  country.  Pupils  who  have  been  taught  the  method  can 
with  difficulty  relinquish  it,  and  if  they  proceed  to  Philosophy 
and  Higher  Machematics  they  will  meet  with  difficulty  in  every 
subject  containing  definitions  involving  ratio.  It  is  proper  to 
remark  that  since  this  article  was  written,  now  some  ten  or 
twelve  years,  several  authors  who  had  adopted  the  new 
method,  have  discarded  it  and  now  use  the  old  method. 


CHAPTER  III. 

NATURE   OF   PROPORTION. 

PROPORTION  arises  from  the  comparison  of  ratios.  Com- 
parison begins  with  comparing  numbers,  giving  rise  to  the 
idea  of  relation,  the  measure  of  which  is  ratio.  After  becoming 
familiar  with  the  idea  of  the  relations  of  numbers,  we  begin  to 
compare  these  relations ;  when  equal  relations  are  compared, 
we  attain  to  the  idea  of  a  Proportion. 

Proportion,  it  is  thus  seen,  has  its  origin  in  comparison;  it  is 
a  comparison  of  the  results  of  two  previous  comparisons.  Every 
proportion  involves  three  comparisons ;  the  two  which  give  rise 
to  the  ratios,  and  a  third,  which  compares  or  equates  the  ratios. 
All  of  these  comparisons  are  exhibited  in  the  expression  of  a 
proportion;  the  symbol  of  ratio  in  the  two  couplets  showing 
the  first  two,  and  the  symbol  of  equality  between  the  couplets 
showing  the  third.  A  proportion,  therefore,  involves  four 
numbers,  so  arranged  that  it  will  appear  that  the  ratio  of  the 
first  to  the  second  equals  the  ratio  of  the  third  to  the  fourth. 
Thus,  the  ratio  of  6  to  3  being  the  same  as  the  ratio  of  8  to  4, 
if  they  are  formally  compared,  as  6  :  3=8  :  4,  we  have  a  pro- 
portion. 

Notation. — A  proportion  may  be  written  by  placing  the  sign 
of  equality  between  the  two  ratios  compared ;  thus  2  :  4=3  :  6. 
Instead  of  the  sign  of  equality,  the  double  colon  is  generally 
used  to  express  the  equality  of  ratios,  the  proportion  being 
written,  2  :  4  : :  3  :  6.  The  symbol  of  equality,  however,  is 
frequently  used  by  the  French  and  German  mathematicians, 
and  is  always  to  be  preferred  in  presenting  the  subject  to 
20  (305) 


306  THE   PHILOSOPHY    OF   ARITHMETIC. 

learners.  A  proportion  may  be  read  in  several  different  ways. 
Thus  we  may  read  the  above  proportion, — "the  ratio  of  2  to  4 
equals  the  ratio  of  3  to  6 ;"  or  "  2  is  to  4  as  3  is  to  6."  The 
latter  is  the  method  generally  used. 

Definition. — A  Proportion  is  the  comparison  of  two  equal 
ratios ;  or,  it  is  the  expression  of  the  equality  of  equal  ratios. 
In  this  expression  the  numbers  that  are  compared  to  obtain  the 
ratio  must  be  indicated.  A  proportion  is  thus  seen  to  be  an 
equation,  and  should  be  thus  regarded.  An  equation,  as  gen- 
erally used,  expresses  the  relation  of  equal  numbers ;  a  pro- 
portion expresses  the  relation  of  equal  ratios.  One  arises  from 
the  comparison  of  quantities ;  the  other,  from  the  comparison 
of  the  relations  of  quantities.  The  former  is  an  equation 
between  equal  numbers ;  the  latter  is  an  equation  between  equal 
ratios. 

The  definition  of  proportion  generally  given  is,  "A  propor- 
tion is  an  equality  of  ratios."  This  is  true,  but  it  is  not  suf- 
ficiently definite  to  constitute  a  perfect  definition.  There  must 
be  not  only  an  equality  of  ratios,  but  a  formal  comparison  of 
these  ratios,  to  produce  a  proportion.  This  comparison  must 
also  exhibit  the  numbers  which  were  compared  to  produce  the 
equal  ratios.  Thus,  the  ratio  of  6  to  3  is  2,  and  the  ratio  of  8 
to  4  is  2  ;  here  is  an  equality  of  ratios,  but  not  a  proportion. 
Again,  if  we  compare  the  ratios  2  and  2,  we  have  the  equation 
2=2,  which  is  not  a  proportion,  since  it  does  not  exhibit  the 
numbers  which  produce  the  equal  ratios.  To  give  a  proportion, 
it  is  essential  that  the  ratios  be  compared,  and  that  the  com- 
parison of  the  numbers  which  give  the  ratios  be  exhibited. 
The  mere  equating  of  the  ratios  is  not  sufficient;  the  propor- 
tion must  show  the  numbers  which,  compared,  give  rise  to 
the  equal  ratios.  A  proportion,  then,  is  not  only  an  "  equality 
of  ratios,"  but  it  is  a  comparison  of  equal  ratios,  in  which 
the  comparison  of  the  numbers  compared  for  a  ratio  is  ex- 
hibited. 

This  idea  of  the  exhibition  of  the  numbers  compared  for  the 


NATURE   OF   PROPORTION.  307 

ratios,  though  not  formally  stated  in  the  definition  which  1 
have  presented,  may  be  directly  inferred  from  it.     For,  if  we 
compare  as  above,  2=2,  so  far  as  we  can  see,  it  is  merely  a 
comparison  of  numbers,  and  not  a  comparison  of  ratios.     It  is 
true  that  every  ratio  is  a  number,  but  the  converse  is  not  true; 
hence  2=2  may  or  may  not  be  the  comparison  of  two  ratios. 
Such  comparison  would  be  indefinite;   therefore,  to  express 
definitely  and  clearly  the  equality  of  ratios,  we  must  retain  the 
numbers  compared,  to  show  that  the  equation  is  an  expression 
of  equal  ratios,  and  not  a  mere  comparison  of  numbers.     The 
definition  is  consequently  regarded  as  sufficiently  explicit  to 
prevent  any  misapprehension.     Should  we  wish  to  incorporate 
this  idea  in  the  definition,  we  might  define  as  follows:    A  Pro- 
portion is  a  comparison  of  equal  ratios,  in  which  the  numbers 
producing  the  ratios  are  exhibited. 

Kinds  of  Proportion.— There  are  several  kinds  of  propor- 
tion, resulting  from  a  modification  or  extension  of  the  pri- 
mary ideas  of  ratio  and  proportion.  A  comparison  of  three  or 
more  pairs  of  numbers  having  equal  ratios,  is  called  Continued 
Proportion.  An  expression  of  the  equality  of  compound  ratios 
is  called  Compound  Proportion.  An  Inverse  Proportion 
is  one  in  which  two  quantities  are  to  each  other  inversely  as 
two  other  quantities.  An  Earmonical  Proportion  is  one  in 
which  the  first  term  is  to  the  last  as  the  difference  between  the 
first  and  second  is  to  the  difference  between  the  last  and  the 
one  preceding  the  last.  We  have  also  Partitive  and  Medial 
Proportion,  which  will  be  defined  subsequently.  The  propor- 
tion requiring  special  consideration  is  Simple  Proportion,  or 
the  comparison  of  two  simple  ratios. 

Principles.— The  principles  of  Proportion  are  the  truths 
which  belong  to  it,  and  which  exhibit  the  relations  between  the 
different  members.  The  fundamental  principle  of  Proportion 
is  that  the  product  of  the  means  equals  the  product  of  the  ex- 
tremes. From  this  we  derive  several  other  principles  by  which 
we  can  find  the  value  of  either  of  the  four  terms  when  the 


308  THE   PHILOSOPHY   OF   ARITHMETIC. 

other  three  are  given.  There  are  many  other  beautiful  princi- 
ples of  Proportion,  besides  this  fundamental  one  and  its  imme- 
diate derivatives,  which  are  not  usually  presented  in  arithmetic, 
but  may  be  found  in  works  on  algebra  and  geometry.  They 
are,  however,  just  as  much  an  essential  part  of  pure  arithmetic 
as  of  geometry,  and  can  all  be  demonstrated  as  easily  here  as 
there.  Indeed,  they  belong  to  arithmetic  rather  than  to  geom- 
etry, since  a  ratio  is  essentially  numerical,  and  hence  should  be 
treated  in  the  science  of  numbers.  These  principles,  it  will  be 
seen,  are  not  self-evident ;  they  admit  of  demonstration.  Re- 
membering this,  it  may  be  asked,  what  then  becomes  of  the 
assertion  of  the  metaphysicians,  that  there  is  no  reasoning  in 
pure  arithmetic  ? 

Demonstration. — The  fundamental  principle  of  Proportion 
may  be  demonstrated  in  two  ways.  The  method  generally 
given  is  the  following :  Take  the  proportion  4:2::  6  :  3. 
From  this  we  have  f=f ;  clearing  of  fractions,  we  have  4x3 
=2x6;  and,  since  4  and  3  are  the  extremes,  and  2  and  6  the 
means,  we  infer  that  the  product  of  the  extremes  equals  the 
product  of  the  means.  This  is  the  method  generally  used  in 
algebra  and  geometry.  Although  entirely  satisfactory  as  a 
demonstration,  the  objection  might  be  made  that  though  it 
proves  that  the  products  are  equal,  it  does  not  show  why  they 
are  equal. 

Another  method  which,  in  arithmetic,  is  preferred  to  the  above, 
is  as  follows :  From  the  fundamental  idea  of  ratio  and  propor- 
tion, we  see  that  in  every  proportion  we  have  2d  term  x  ratio 
.  2d  term  : :  4th  term  x  ratio  :  4th  term.  Now,  in  the  product 
of  the  extremes,  we  have  2d  term,  ratio,  and  4th  term,  and  in 
the  product  of  the  means,  we  have  the  same  factors ;  hence 
the  products  are  equal.  This  is  a  simple  method,  clearly  seen, 
and  shows  not  only  that  the  products  are  equal,  but  that  they 
must  be  so,  and  why  they  are  so,  which  the  other  method  does 
not.  The  products  are  seen  to  be  equal  because  in  the  very 
nature  of  the  subject  they  contain  the  same  factors. 


NATURE  OF    PROPORTION.  309 

The  same  demonstration  may  be  put  in  the  more  concise 
language  of  algebra.  Take  the  proportion  a  :  b  : :  c  :  d,  let  r 
=  the  ratio,  then  we  have  a^-b=r,  hence  a=b.r,  and  in  the 
same  way  c=d.r ;  hence  the  proportion  becomes  b.r  :  b  : :  d.r 
:  d.  Now,  in  the  extremes  we  have  b,  r,  and  d,  and  in  the 
means  we  have  the  same  factors  ;  hence  the  two  products  will 
be  equal. 


CHAPTER  IV. 

APPLICATION  OP  SIMPLE   PROPORTION. 

SIMPLE  PROPORTION  is  employed  in  the  solution  of  prob- 
lems in  which  three  of  four  quantities  are  given,  to  find  the 
fourth.  These  quantities  must  be  so  related  that  the  required 
quantity  bears  the  same  relation  to  the  given  quantity  of  the 
same  kind  that  one  of  the  two  remaining  quantities  does  to  the 
other.  We  can  then  form  a  proportion  in  which  one  term  is 
unknown,  and  this  unknown  term  can  be  found  by  the  principles 
of  proportion.  Thus,  suppose  the  problem  to  be, — What  cost 
3  yards  of  cloth,  if  2  yards  cost  $8  ? 

Here  we  see  that  the  operation. 

cost  of  3  yards  bears  the     Cost  of  3  yds.  :  $8  : :  3  yds.  :  2  yds ; 
same  relation  to  the  cost     Cost  of  3  yds .=-—=$12. 
of  2  yards  that  3  yards 

bears  to  2  yards ;  nence  we  have  the  proportion  given  in  the 
margin,  from  which  we  readily  find  the  value  of  the  unknown 
term. 

In  all  such  problems  three  terms  are  given  to  find  the  fourth ; 
from  which  Simple  Proportion  has  been  called  the  Rule  of 
Three.  It  was  regarded  as  very  important  by  the  old  school 
of  arithmeticians,  and  was  by  them  called  "The  golden  rule  of 
three."  It  is  now  falling  into  disrepute,  the  beautiful  system 
of  analysis  having,  to  a  great  extent,  taken  its  place.  The 
method  of  analysis  is  simpler  in  thought  than  that  of  proportion, 
and  in  many  cases  is  to  be  preferred  to  the  solution  by  propor- 
tion, especially  in  elementary  arithmetic;  but  still  the  rule  of 

(310) 


APPLICATION  OF   SIMPLE   PROPORTION.  311 

Simple  Proportion  should  not  be  entirely  discarded.  The 
comparison  of  elements  by  proportion  affords  a  valuable  disci- 
pline and  should  be  retained  for  educational  reasons;  and 
moreover  it  is  also  valuable,  if  not  indispensable,  in  the  solu- 
tion of  some  problems  which  can  hardly  be  reached  by  analysis. 
In  algebra,  geometry,  and  the  higher  mathematics,  it  is,  of 
course,  indispensable. 

Position  of  the  Unknown  Quantity. — It  is  seen  that,  in  the 
solution  of  the  preceding  problem  by  proportion,  I  place  the 
unknown  quantity  in  the  first  term.  This  is  not  in  accordance 
with  general  custom;  other  writers  place  the  unknown  quan- 
tity in  the  fourth  term.  I  have  ventured  to  depart  from  this 
custom,  and  to  recommend  the  general  adoption  of  such  a  depar- 
ture, for  reasons  which  seem  to  me  conclusive.  These  reasons 
are  twofold:  first,  the  method  suggested  is  dictated  by  the 
laws  of  logic;  and,  second,  it  is  more  convenient  in  practice. 
Both  of  these  points  will  be  briefly  considered. 

First.  The  law  of  correct  reasoning  is  to  compare  the  unknown 
with  the  known,  not  the  known  with  the  unknown.  The  ordi- 
nary method  begins  the  proportion  with  the  known  quantities, 
thus  comparing  the  known  with  the  unknown,  in  violation  of 
an  established  principle  of  logic.  The  method  I  have  suggested 
commences  with  the  unknown  quantity,  and  thus  compares  the 
unknown  with  the  known,  in  conformity  to  the  laws  of  thought. 
It  seems  therefore  that  the  old  method  is  not  logically  accu- 
rate, and  that  the  correct  method  of  solving  a  problem  in  Rule 
of  Three  is  to  place  the  unknown  quantity  in  the  first  term. 

Second.  The  method  proposed  will  be  found  to  be  much 
more  convenient  in  practice.  A  proportion  is  more  easily 
stated  by  beginning  it  with  the  unknown  term.  This  will 
be  especially  appreciated  by  those  who  have  taught  Trigo- 
nometry. In  stating  a  proportion  so  as  to  get  the  required 
quantity  in  the  last  term,  I  have  seen  pupils  try  two  or  three 
statements  before  obtaining  the  right  one.  It  cannot  be  readily 
seen  how  the  proportion  should  begin  so  that  the  unknown 


312  THE    PHILOSOPHY    OF    ARITHMETIC. 

quantity  shall  come  in  the  last  term.  If,  however,  the  pupil 
begins  the  proportion  with  that  which  he  wishes  to  find,  the 
other  terms  will  arrange  themselves  without  any  difficulty. 
Suppose,  for  instance,  that  we  wish  to  obtain  an  unknown 
angle  of  a  triangle.  If  we  reason  thus :  sine  of  the  required 
angle  is  to  the  sine  of  the  given  angle  as  the  side  opposite  the 
required  angle  is  to  the  side  opposite  the  given  angle;  the 
pupil  will  write  the  proportion  without  any  hesitation.  If  we 
reverse  this  order,  it  is  necessary  to  go  through  the  whole 
comparison  mentally  before  beginning  to  write,  so  that  we 
may  be  sure  to  close  the  proportion  with  the  required  quantity. 
It  is  therefore  believed  that  the  simplest  method  of  stating  a 
proportion  is  to  place  the  unknown  quantity  in  the  first  term. 

The  utility  of  this  change  has  been  frequently  illustrated  in 
my  own  experience.  I  remember,  while  visiting  a  young 
women's  college,  hearing  a  recitation  in  geometry  in  which  the 
professor  was  trying  to  lead  a  pupil  to  state  a  proportion  from 
which  a  certain  line  could  be  determined.  The  young  lady  made 
several  attempts  and  failed,  when  I  said,  "Professor,  let  her 
begin  with  the  line  she  wishes  to  find."  He  accepted  the  sug- 
gestion, and  she  immediately  stated  the  proportion  correctly. 

Several  authors  suggest  that  the  unknown  quantity  should 
be  placed  sometimes  in  one  term  and  sometimes  in  another  to 
test  the  pupil's  knowledge  of  the  subject.  This  is  a  valuable 
suggestion  ;  but  any  position  of  the  unknown  term  except  in 
the  fourth  term  they  regard  not  as  a  general,  but  as  an  excep- 
tional method.  Their  rule  is  to  place  the  unknown  term  last ; 
any  other  arrangement  is  the  exception.  What  I  claim  is  that 
the  placing  of  the  unknown  quantity  in  the  first  term  should 
be  the  rule,  and  any  other  arrangement  the  exception.  It  is 
recommended  also  that  the  teacher  require  the  learner  to  place 
it  in  different  terms,  that  he  may  acquire  a  clear  and  complete 
idea  of  the  subject. 

Symbol  for  the  Unknown. — Some  authors  employ  the  letter 
x  in  arithmetic  as  a  symbol  for  the  unknown  quantity.     Thus, 


APPLICATION   OF    SIMPLE    PROPORTION.  313 

in  the  problem  previously  presented,  we  may  write  x  :  $8  :  :  3  : 
2.  This  practice  is  derived  from  the  French,  and  is  commend- 
able. It  is  sometimes  objected,  that  it  is  introducing  algebra 
into  arithmetic  ;  but  such  objection,  however,  is  not  valid.  Al- 
gebra and  arithmetic  are  not  two  distinct  sciences,  but  rather 
branches  of  the  same  science.  The  former,  at  least  in  its  ele- 
ments, is  but  a  more  general  kind  of  arithmetic ;  and  it  is  not 
at  all  improper  to  introduce  its  concise  and  general  language 
into  arithmetic.  I  think  it  well,  with  younger  pupils,  to  ex- 
press the  unknown  term  in  an  abbreviated  form  as  is  indicated 
in  the  previous  solution;  when  pupils  become  familiar  with 
this,  I  would  use  the  symbol  x  as  a  representative  of  it. 

Three  Terms  Statement. — It  is  seen  that  in  the  solution  of 
the  given  problem  in  proportion,  I  use  four  terms  in  the  state- 
ment. Many  authors,  however,  use  only  three  terms  in  stating 
a  proportion.  This  was  the  method  of  the  old  authors,  when 
rules  reigned  and  principles  were  ignored,  in  what  might  be 
called  "the  dark  ages"  of  arithmetic.  Several  recent  writers 
have  broken  away  from  the  old  usage,  and  write  the  proportion 
with  four  terms  instead  of  three.  It  is  unnecessary  to  say 
that  the  old  method  was  incomplete  and  incorrect.  An  ex- 
pression is  not  a  proportion  unless  it  has  four  terms.  The  old 
method  was  merely  mechanical,  and  gave  the  pupil  no  idea,  or 
at  least  a  very  imperfect  idea,  of  the  true  nature  of  proportion. 
The  sooner  the  new  method  is  generally  adopted  the  better  for 
science  and  education. 

Method  of  Statement. — No  subject  in  arithmetic  is  so  illogi- 
cally  presented  as  Simple  Proportion  in  its  application  to  the 
solution  of  problems.  In  the  statement  of  the  proportion,  all 
reasoning  seems  to  be  completely  ignored,  and  the  whole  thing 
becomes  a  mere  mechanical  operation  for  the  answer.  The  pro- 
cess is  as  follows :  "  Write  that  number  which  is  like  the  answer 
sought  as  the  third  term ;  then  if  the  answer  is  to  be  greater 
than  the  third  term,  make  the  greater  of  the  two  remaining 
numbers  the  second  term  and  the  smaller  the  first  term,"  etc. 
14 


314  THE    PHILOSOPHY    OF    ARITHMETIC. 

Now,  though  this  might  do  well  enough  as  a  rule  for  get 
ting  an  answer,  to  require  the  pupils  to  explain  the  solution  by 
it,  as  is  done  in  many  instances,  is  to  rob  the  subject  of  any 
claims  to  a  scientific  process.  The  pupil  thus  taught  to  solve 
his  problems  has  no  more  idea  of  proportion  than  if  the  subject 
were  not  presented  in  the  book.  The  whole  process  becomes  a 
piece  of  charlatanism,  utterly  devoid  of  all  claims  to  science. 
A  better  rule  would  be  this  :  Write  the  number  like  the  answer ; 
if  the  answer  is  to  be  greater,  multiply  by  the  greater  of  the 
other  two  numbers  and  divide  by  the  less,  etc.  This  would  be 
the  better  method,  since  it  makes  no  claims  to  be  a  scientific 
process,  as  the  other  does.  Both  methods  are  absurd  as  a  pro- 
cess of  reasoning  in  Arithmetic ;  but  the  latter  less  so,  since  it 
makes  no  pretensions  to  be  a  reasoning  process. 

What  then  is  the  true  method  ?  I  answer,  if  a  pupil  cannot 
state  a  proportion  by  actual  comparison  of  the  elements  of  the 
problem,  he  is  not  prepared  for  proportion,  and  should  solve 
the  question  by  analysis.  If  he  uses  proportion,  he  should  use 
it  as  a  logical  process  of  reasoning,  and  not  as  a  blind  mechan- 
ical form  to  get  the  answer.  He  should  then  be  required  to 
reason  thus:  Since  the  cost  of  3  yds.  bears  the  same  relation  to 
the  cost  of  2  yds.  that  3  yds.  bear  to  2  yds.,  we  have  the  pro- 
portion, cost  of  3  yds.  :  $8  : :  3  yds.  :  2  yds. 

If  this  is  not  evident  and  cannot  be  readily  seen,  then  we 
should  dispense  with  proportion  until  the  pupil  is  old  enough 
to  understand  it,  and  require  the  problems  to  be  solved  by  analy- 
sis. If  the  unknown  quantity  be  placed  in  the  last  term  we 
would  reason  thus :  Since  2  yds.  bear  the  same  relation  to  3 
yds.  that  the  cost  of  2  yds.  bears  to  the  cost  of  3  yds,  we  have 
the  proportion,  2  yds.  :  3  yds.  : :  $8  :  cost  of  3  yds. 

Cause  and  Effect. — A  new  method  of  explaining  proportion 
has  recently  been  introduced  into  arithmetic,  which  may  be 
called  the  method  of  Cause  and  Effect.  All  problems  in  pro- 
portion, it  is  said,  may  be  considered  as  a  comparison  of  two 
causes  and  two  effects;   and  since  effects  are  proportional  to 


APPLICATION    OF    SIMPLE    PROPORTION.  315 

causes,  a  problem  is  supposed  to  be  readily  stated  in  a  propor- 
tion. To  illustrate,  take  the  problem,  If  2  horses  eat  6  tons  of 
hay  in  a  year,  how  much  will  3  horses  eat  in  the  same  time  ? 
Here  the  horses  are  regarded  as  a  cause  and  the  tons  of  hay  as 
an  effect,  and  the  reasoning  is  as  follows:  2  horses  as  a  cause 
bear  the  same  relation  to  3  horses  a,s  a  cause,  that  6  tons  as  an 
effect,  bears  to  the  required  effect ;  from  which  we  have  a  pro- 
portion and  can  determine  the  required  term. 

This  method  was  first  introduced  into  arithmetic  by  Prof. 
H.  iST.  Robinson,  and  has  been  adopted  by  several  authors. 
The  same  idea  was  presented  by  an  arithmetician  of  Verona, 
who  distinguished  the  quantities  into  agents  and  patients.  It 
is  supposed  that  it  tends  to  simplify  the  subject,  enabling 
learners  more  readily  to  state  a  proportion  than  by  a  simple 
comparison  of  the  elements.  This  supposition,  however,  is  not 
founded  in  truth.  Instead  of  simplifying  the  subject,  the  method 
of  cause  and  effect  really  increases  the  difficulty  and  tends  to 
confuse  the  mind.  It  lugs  into  arithmetic  an  idea  foreign  to 
the  subject,  to  explain  relations  which  are  much  more  evident 
than  the  relation  of  cause  and  effect. 

Another  objection  to  the  method  is  that  the  relation  of  quan- 
tities as  cause  and  effect  is  often  rather  fancied  than  real.  In 
many  cases,  indeed,  there  is  no  such  relation  existing  at  all. 
Take  the  problem,  "If  a  man  walks  6  miles  in  2  hours,  how  far 
will  he  walk  in  5  hours  ?"  Will  the  pupil  readily  see  which 
is  the  cause  and  which  the  effect  ?  Will  the  advocates  of  the 
method,  tell  us  whether  the  6  miles  or  the  2  hours  are  to  be 
regarded  as  the  cause?  Or  take  the  problem,  "If  18d.  ster- 
ling equal  36  cts.  U.  S.,  what  are  54d.  sterling  worth?'' 
Would  not  the  pupils  be  puzzled  to  tell  which  is  the  cause  and 
which  the  effect?  Indeed,  there  is  no  relation  of  cause  and 
effect  in  a  large  number  of  such  problems  ;  and  any  effort  to 
establish  such  a  relation  will  confuse  that  which  is  simple  and 
easily  understood. 

If  anything  further  is  needed  to  show  the  incorrectness  of 


316  THE    PHILOSOPHY   OF    ARITHMETIC. 

the  method,  take  a  problem  in  what  is  called  Inverse  Proportion. 
Thus,  "If  3  men  do  a  piece  of  work  in  8  days,  in  what  time  will 
6  men  do  it?"  Here  3  men  and  8  days  would  be  regarded  as 
the  first  cause  and  effect,  and  6  men  and  the  corresponding 
number  of  days  as  the  second  cause  and  effect.  Now,  if  we 
form  a  proportion,  we  have  the  first  cause  is  to  second  cause  as 
the  second  effect  is  to  the  first  effect ;  from  which  we  see  that 
in  this  case  like  causes  are  not  to  each  other  as  like  effects,  a 
conclusion  which  completely  contradicts  the  fundamental  prin- 
ciple of  the  relation  of  cause  and  effect. 

Inverse  Proportion. — There  is  a  class  of  problems  which  give 
rise  to  what  is  called  Inverse  Proportion.  In  this  the  two 
quantities  of  the  same  kind  are  to  each  other,  not  directly  as  the 
other  two  quantities  in  the  order  of  their  relation,  but  rather 
inversely  as  those  quantities.  Thus,  in  the  problem,  "If  3 
men  build  a  fence  in  12  days,  in  what  time  will  9  men  build 
it?"  Here  we  have  the  required  time  is  to  12  days,  not  as  9 
men  to  3  men,  but  as  3  men  to  9  men ;  that  is,  inversely  as 
the  order  indicated  by  the  order  of  the  terms  of  the  first  couplet. 
This  is  sometimes  called  Reciprocal  Proportion,  since  the  quan- 
tities are  as  the  reciprocals  of  9  and  3 ;  that  is  as  \  to  ^  or  3  to  9 

Many  problems  in  Inverse  Proportion  may,  however,  be 
stated  in  a  direct  proportion.  To  illustrate,  take  the  problem 
just  solved.  Now,  if  3  men  do  a  piece  of  work  in  12  days,  in 
1  day  they  will  do  -^  of  it,  and  if  a  number  of  men  do  a  piece 
of  work  in  4  days,  in  1  day  they  will  do  £  of  it ;  hence,  since 
the  number  of  men  are  to  each  other  as  the  work  done,  we  have 
the  direct  proportion,  "the  number  of  men  required  is  to  3  men, 
as  ^  to  -J^,"  from  which  we  can  readily  find  the  term  required. 
If,  in  this  proportion,  we  multiply  the  second  couplet  by  48,  it 
will  become  12  :  4,  which  gives  the  same  proportion  as  that 
which  was  obtained  by  the  method  of  inverse  proportion.  It 
is  thus  seen  that,  in  some  cases  at  least,  the  method  of  inverse 
proportion  may  be  avoided,  and  the  problem  be  expressed  by  a 
direct  proportion. 


APPIJCATION   OF   SIMPLE    PROPORTION.  317 

If,  however,  in  the  above  problem  the  number  of  men  in 
both  cases  had  been  given,  and  the  number  of  days  in  one  case 
required,  the  problem  could  not  be  conveniently  stated  in  a 
direct  proportion,  since  to  do  so  would  require  the  reciprocal 
of  the  unknown  quantity.  Should  this  quantity  be  represented 
by  an  algebraic  symbol,  however,  we  could  still  state  the  pro- 
portion directly,  and  readily  find  the  unknown  quantity. 

Proportion  distinctly  Arithmetical— -The  subject  of  propor- 
tion  is  purely  an  arithmetical  process.      Ratio  is  a  number, 
hence  proportion,  arising  from  the  comparison  of  ratios,  must 
be  numerical.     These  ratios   may  arise  from  comparing  con- 
tinuous or  discrete  quantities,  hence  we  may  have  a  propor- 
tion wherein  geometrical  quantities  are  compared.     Attention 
is  called  to  the  fact,  however,  that  the  principles  of  proportion 
are  only  generally  true  with  respect  of  numbers.     A  propor- 
tion in  geometry,  comparing  four  surfaces  or  volumes,  may  be 
true,  but  the  principles  of  a  proportion  can  have  no  meaning  in 
such  a  case.     In  taking  the  product  of  the  means  equal  to  the 
product  of  the  extremes,  we  shall  have  one  surface  or  one  vol- 
ume multiplied  by  another,  which  can  mean  nothing  unless 
they  be  regarded  as  numbers.      In  geometry  we  regard  the 
product  of  two  lines  as  giving  a  surface,  and  the  product  of  a 
line  and  surface  as  giving  a  volume ;  but  what  idea  can  we 
attach  to  the  product  of  two  surfaces  or  two  volumes  ?     It  is 
thus  seen  that  Proportion  is  essentially  a  process  of  numbers, 
and  is,  therefore,  a  branch  of  Pure   Arithmetic.     Since   the 
principles  of   Proportion  admit  of  demonstration,  we  inquire 
again  what  becomes  of  MansePs  assertion  that  "  Pure  Arith- 
metic  contains  no  demonstration  ?" 


CHAPTER  V. 
COMPOUND   PROPORTION. 

A  COMPOUND  PROPORTION  is  a  proportion  in  which  one 
or  both  ratios  are  compound.  It  is  employed  in  the  solu- 
tion of  problems  in  which  the  required  term  depends  upon  the 
comparison  of  more  than  two  elements.  In  Simple  Proportion 
the  unknown  quantity  depends  upon  a  comparison  of  two  ele- 
ments forming  one  pair  of  similar  quantities;  in  Compound 
Proportion  it  depends  upon  the  comparison  of  several  elements 
forming  two  or  more  pairs  of  similar  quantities. 

A  Compound  Ratio  has  been  denned  as  the  product  of  two 
or  more  simple  ratios.     The  expression  of  a  compound  ratio  is 

i  \  •  in!  *     If  su°k  a  ratio  be  comPare(*  to  an  e(lual  simple 
ratio,  or  if  two  such  compound  ratios  be  compared  with  each 

2  ;  g  >  : :  7  :  56 

and  {?  !  in}  :  :  I?  •  14 1  are  examPles  of  compound  propor- 
tion. In  these  expressions  we  mean  that  the  value  of  the  first 
couplet  equals  the  value  of  the  second;  thus,  in  the  first  pro- 
portion we  have  fxf  or  ■&  equals  -^g-;  in  the  second,  f  XTV= 

The  subject  of  Compound  Proportion  has  been  even  more 
unscientifically  treated,  if  possible,  than  Simple  Proportion.  In 
no  work  upon  Arithmetic,  and  indeed  in  no  work  upon  Algebra, 
have  I  seen  the  subject  presented  in  a  really  scientific  manner. 
As  a  general  thing,  problems  are  given  under  the  head  of  com- 
oound  proportion,  to  be  solved  either  mechanically  by  rule,  or 
else  by  analysis,  which,  of  course,  is  not  compound  proportion. 

(318) 


COMPOUND   PROPORTION.  319 

The  principles  of  a  compound  proportion  are  not  developed, 
and  in  its  application  it  is  regarded,  not  as  a  scientific  process, 
but  as  a  machine  for  working  out  the  answer.  This,  of  course, 
is  not  as  it  should  be.  Compound  Proportion  is  just  as  much 
a  scientific  process  as  Simple  Proportion,  and  demands  just  as 
logical  a  treatment.  I  will  enforce  what  I  mean  by  calling 
attention  to  a  few  of  the  principles  of  such  a  proportion,  and 
then  showing  its  scientific  application. 

Principles. — In  Compound  Proportion  we  have  certain  defi- 
nite scientific  principles,  as  in  Simple  Proportion.  A  few  of 
these  principles  will  now  be  stated. 

1.  The  product  of  all  the  terms  in  the  means  equals  the  pro- 
duct of  all  the  terms  in  the  extremes.  To  show  the  truth  of 
this,  take  the  proportion,  given 

in  the  margin.     From  the  prin-  operation. 

ciples  of  compound   ratio   we         "]  5  -  10  k    ::    "3  7  •  14  t 

have  f  x  A=*X-ft;  and  clear.  *2X_5_=3X_^ 

ing  this  of  fractions  we  have     2x5x6x14=3x7x4x10. 

2x5x6xl4  =  3x  7x4x10, 

which,  by  examining  the  terms,  we  see  proves  the  principle. 

From  this  principle  we  can  immediately  derive  two  others. 

2.  Any  term  in  either  extreme  equals  the  product  of  the 
means,  divided  by  the  product  of  the  other  terms  in  the  ex- 
tremes. 

3.  Any  term  in  either  mean  equals  the  product  of  the  extremes 
divided  by  the  product  of  the  other  terms  in  the  means. 

Other  principles  can  also  be  derived,  as  in  Simple  Proportion, 
but  the  three  given  are  all  that  are  necessary  in  arithmetic. 

Application. — In  the  application  of  Compound  Proportion 
to  the  solution  of  problems,  we  should  proceed  upon  the  same 
principles  of  comparison  employed  in  Simple  Proportion.  If 
we  do  not,  the  process  is  not  Compound  Proportion,  and  should 
not  be  so  regarded.  To  illustrate  the  true  method,  we  take  the 
problem,  "If  4  men  earn  $24  in  7  days,  how  much  can  14  men 
earn  in  12  days?" 


320  THE    PHILOSOPHY    OP   ARITHMETIC. 

In  the  solution  of  this  problem  by  Compound  Proportion,  we 
should  reason  thus :     The  sum  earned  is  in  proportion  to  the 
number  of  men  and  the  time  they  labor;  hence  the  sum  14  men 
can  earn  is  to  $24,  the  sum  that  4  men 
earn,  as  14  men  to  4  men,  and  also  as  operation. 

12  days  to  7  days;   giving  the  com-     Sum  :  24  :  :    -j  r*  ;  *1 
pound  proportion  which  is  presented  24x14x12 

in  the  margin.     From  this  we  find  the         Sum=       j— ^ 
unknown  term  to  be  $144.     Or  we  may 

enter  a  little  more  into  detail,  and  say — The  sum  14  men  can 
earn  in  7  days  is  to  the  sum  4  men  can  earn  in  7  days,  as  14  men 
is  to  4  men;  and  also  the  sum  14  men  can  earn  in  12  days  is 
to  the  sum  that  they  can  earn  in  7  days,  as  12  is  to  7 ;  hence 
we  have  the  compound  proportion  given  in  the  margin. 

By  Analysis. — The  subject  of  Compound  Proportion  is  some- 
what difficult,  in  fact  too  difficult,  for  young  students  in  arith- 
metic. With  such  the  method  of  analysis  should  be  used 
instead  of  proportion.  The  analytical  method  is  clear  and 
simple,  and  will  be  readily  understood.  It  should  be  borne  in 
mind,  however,  that  when  we  solve  by  analysis  we  are  not 
solving  by  compound  proportion,  a  fact  that  seems  sometimes 
to  be  forgotten. 

In  solving  the  preceding  problems  by  analysis,  it  is  necessary 
to  pass  from  the  4  men  to  14  men,  and  from  the  7  days  to  12 
days,  the  sum  earned  varying  as  we  make  the  transposition :  to 
do  this  we  pass  from  the  collection  to  the  unit,  and  then  from 
the  unit  to  the  collection.  The  solution  is  as  follows,  the  work 
being  as  indicated  in  the  margin. 

If  4  men  earn  $24  in  7  days  one  man  will  earn  £  of  $24,  and 
14  men  will  earn  14  times  £  or  ^  of  operation. 

$24.    If  14  men  earn  J^x  $24  in  7  days,     sum==i_2x iAx$24. 
in  one  day  they  will  earn  |  of  ±£  of  $24, 

and  in  12  days  they  will  earn  12  times  |  of  *£■  of  $24,  which  is 
\f-  of  i£  of  $24,  which  by  cancelling,  we  find  equals  $144.  In- 
stead of  putting  it  in  the  form  of  a  compound  fraction,  we  could 


PARTITIVE    PROPORTION.  321 

have  made  the  reduction  as  we  passed  along ;  but  in  compli- 
cated problems  the  method  here  used  is  preferred,  as  the  can- 
cellation of  equal  factors  will  often  greatly  abridge  the  process. 

PARTITIVE   PROPORTION. 

The  subject  of  ratio  gives  rise  to  several  arithmetical 
processes  which  have  received  the  name  of  Proportion. 
Among  these  we  have  Partitive  Proportion,  Conjoined  Pro- 
portion, Medial  Proportion,  Geometrical  Proportion,  etc.  Geo- 
metrical Proportion  embraces  Simple  Proportion,  Compound 
Proportion,  Inverse  Proportion,  etc.  The  other  kinds  are 
distinguished  by  their  special  names.  When  we  speak  of  pro- 
portion, without  any  qualifying  word,  we  mean  Geometrical 
Proportion.  Geometrical  Proportion  has  been  treated  in  the 
preceding  part  of  this  chapter ;  the  other  varieties  of  proportion 
will  now  be  presented. 

The  comparison  of  numbers  gives  rise  to  a  division  of  them 
into  parts  which  shall  bear  a  given  relation  to  each  other.  This 
process  has  received  the  name  of  Partitive  Proportion.  Parti- 
tive Proportion  is  the  process  of  dividing  numbers  into  parts 
bearing  certain  relations  to  each  other.  To  illustrate,  suppose 
it  be  required  to  divide  24  into  two  parts,  one  of  which  is  twice 
the  other.  An  equivalent  problem  is,  "  Given  the  sum  of  two 
numbers  equal  to  24,  and  one  of  the  numbers  twice  the  other ; 
what  are  the  numbers  ?" 

Origin. — Partitive  Proportion  is  a  process  of  pure  arithme- 
tic ;  it  originated,  however,  in  the  application  of  numbers  to 
business  transactions.  Partnership  is  a  case  of  Partitive  Pro- 
portion. But,  although  the  subject  had  its  origin  in  the  appli- 
cation of  numbers,  it  is  now,  in  accordance  with  the  law  of  the 
growth  of  science,  a  purely  abstract  process. 

Cases. — This  subject  embraces  quite  a  large  number  of  cases, 

arising  from  the  various  relations  that  may  exist  among  the 

several  parts  into  which  a  number  is  divided.     It  is  evident, 

also,  that  the  greater  the  number  of  the  parts  the  more  compli- 

21 


322  THE    PHILOSOPHY    OF    ARITHMETIC. 

cated  will  become  the  process.     The  most  important  cases  are 
the  following: 

1.  When  the  parts  are  all  equal. 

2.  When  one  part  is  a  number  more  or  less  than  the  other. 

3.  When  one  part  is  a  number  of  times  the  other. 

4.  When  one  part  is  a  fractional  part  of  the  other. 

5.  When  the  parts  are  to  each  other  as  given  integers. 

6.  When  the  parts  are  to  each  other  as  given  fractions. 

7.  When  a  number  of  times  one  part  equals  a  number  of 
times  another. 

8.  When  a  fractional  part  of  one  equals  a  fractional  part  of 
another. 

These  simple  cases,  it  is  evident,  may  be  combined  with  each 
other,  giving  rise  to  others  more  complicated  than  any  of  these. 
A  little  ingenuity  will  suggest  a  large  number  of  such  cases, 
some  of  which  will  be  quite  interesting. 

Method  of  Treatment. — To  illustrate  the  character  of  one  of 
the  simple  cases  and  its  treatment,  let  us  take  a  problem  and 
its  solution.  Case  8  will  give  us  a  problem  like  the  following: 
Divide  34  into  two  parts  such  that  §  of  the  first  part  equals  f 
of  the  second  part.  The  solution  of  this  case  is  as  follows : 
If  |  of  the  first  equals  f  of  the  second,  £  of  the  first  equals  ^  of 
|  or  f  of  the  second,  and  |  of  the  first  equals  f  of  the  second ; 
then  |  of  the  second,  which  is  the  first,  plus  |  of  the  second,  or 
ig£  of  the  second  part,  equals  34,  etc.  The  other  cases  are  solved 
in  my  Mental  Arithmetic,  and  need  not  be  presented  here. 

CONJOINED   PROPORTION. 

The  comparison  of  numbers  also  gives  rise  to  an  arithmetical 
process  which  has  received  the  name  of  Conjoined  Proportion. 
Conjoined  Proportion  is  the  process  of  comparing  terms  so 
related  that  each  consequent  is  of  the  same  kind  as  the  next 
antecedent.  The  character  of  the  subject  is  seen  by  the  follow- 
ing concrete  problem:  "What  cost  8  apples,  if  4  apples  are 
worth  2  oranges,  and  3  oranges  are  worth  6  melons,  and  4 
melons  are  worth  12  cents?" 


CONJOINED  PROPORTION.  323 

An  abstract  problem,  showing  that  it  is  a  process  of  pnre 
arithmetic,  is  as  follows:  "If  twice  a  number  equals  4  times 
another  number,  and  3  times  the  second  number  equals  6  times 
a  third  number,  and  4  times  the  third  number  equals  2  times  a 
fourth  number,  and  5  times  the  fourth  number  equals  40  ;  what 
is  the  first  number  ?" 

Method  of  Treatment. — Conjoined  Proportion  is  treated  by 
analysis,  and  presents  a  very  interesting  application  of  the 
analytical  method  of  reasoning.  *  The  problems  may  be  solved 
in  two  ways  somewhat  distinct ;  that  is,  we  may  begin  at  the 
latter  part  of  the  problem,  and  work  back,  step  by  step,  to  the 
beginning ;  or  we  may  commence  at  the  beginning  of  the  prob- 
lem and  pass  from  quantity  to  quantity,  in  regular  order,  until 
we  find  the  value  of  the  first  quantity  in  terms  of  the  last.  To 
illustrate,  the  problem  given  may  be  solved  thus : 

Solution  1. — If  5  times  the  fourth  number  equals  40,  once 
the  fourth  number  equals  \  of  40,  or  8,  and  twice  the  4th,  which 
equals  4  times  the  3d,  equals  2  times  8,  or  16.  If  4  times  the 
3d  equals  16,  once  the  3d  equals  \  of  16,  or  4,  and  6  times  the 
3d  or  3  times  the  2d  equals  6  times  4,  or  24 ;  and  so  on  until 
we  reach  once  the  1st  number. 

Solution  2. — If  twice  a  number  equals  4  times  another,  once 
the  number  equals  J  of  4  times,  or  two  times  the  2d ;  if  3  times 
the  2d  equals  6  times  the  3d,  once  the  2d  equals  ^  of  6  times, 
or  2  times  the  3d,  and  2  times  the  2d,  or  the  1st,  equals  twice 
2  times  the  3d,  or  4  times  the  3d ;  and  so  on  until  we  find  once 
the  1st  in  terms  of  the  given  quantity. 

Both  of  these  methods  are  simple  and  logical.  The  first 
method  will  probably  be  preferred  for  its  directness  and  sim- 
plicity. It  may  also  be  remarked  that  these  problems  can  be 
solved  by  Compound  Proportion,  and  perhaps  might  have  been 
logically  treated  under  that  head. 

MEDIAL   PROPORTION. 

The  comparison  of  numbers  and  the  combining  of  them  in 
certain  relations,  give  rise  to  an   arithmetical   process  which 


324  THE   PHILOSOPHY    OF    ARITHMETIC. 

has  received  the  name  of  Medial  Proportion.  Medial  Propor- 
tion is  the  process  of  finding  in  what  ratio  two  or  more  quan- 
tities may  be  combined,  that  the  combination  may  have  a  mean 
or  average  value. 

The  subject,  in  its  application,  is  usually  called  Alligation, 
from  alligo,  I  bind  or  unite  together,  the  name  being  suggested, 
probably,  by  the  method  of  solution,  which  consisted  of  linking 
or  uniting  the  figures  with  a  line.  It  may,  however,  have  been 
suggested  by  the  nature  of  the  process  itself,  in  which  the  sev- 
eral quantities  are  combined. 

Origin. — Medial  Proportion  also  originated  in  the  concrete, 
that  is,  in  the  application  of  numbers.  Indeed,  even  now  it  is 
difficult  to  present  it  as  an  abstract  process  ;  that  is,  as  a  process 
of  pure  number.  It  is  so  intimately  associated  with  the  combi- 
nation of  things  of  different  values,  that  it  is  very  difficult  to 
apply  it  to  the  combination  of  abstract  numbers.  Still  it  is 
evidently  a  process  of  pure  arithmetic ;  and  its  importance  and 
distinctive  character,  even  as  an  application  of  numbers,  lead 
me  to  speak  of  it  in  this  connection. 

Cases. — The  subject  presents  a  number  of  cases,  the  most 
important  of  which  are  the  following: 

1.  Given,  the  quantity  and  value  of  each,  to  find  the  mean 
value. 

2.  Given,  the  mean  value  and  the  value  of  each  quantity,  to 
find  the  proportional  quantity  of  each. 

3.  Given,  the  mean  value,  the  value  of  each,  and  the  relative 
amounts  of  two  or  more,  to  find  the  other  quantities. 

4.  Given  the  mean  value,  the  value  of  each,  and  the  quantity 
of  one  or  more,  to  find  the  other  quantities. 

5.  Given,  the  mean  value,  the  value  of  each,  and  the  entire 
quantity,  to  find  the  quantity  of  each. 

Method  of  Treatment.— As  formerly  treated,  the  subject  was 
one  of  the  most  mechanical  in  arithmetic.  The  old  "linking 
process,"  as  presented  in  the  text-books,  was  seldom  understood 
either  by  teacher  or  pupil.     Recently,  however,  Prof.  Wood, 


MEDIAL   PROPORTION.  325 

formerly  of  che  New  York  State  Normal  School,  has  made  a 
very  happy  application  of  analysis  to  the  solution  of  this  class 
of  problems,  and  poured  a  flood  of  light  upon  the  subject,  so 
that  it  is  now  one  of  the  most  interesting  processes  of  arithmetic. 
It  has  extended  the  domain  of  the  subject  also,  so  that  it  includes 
some  of  the  more  difficult  cases  of  Indeterminate  Analysis,  for 
an  illustration  of  which  see  my  Higher  Arithmetic. 

The  method  of  treatment  is  to  compare  one  number  above 
the  average  with  one  below  it  by  their  relation  to  the  average, 
finding  how  much  must  be  taken  to  gain  or  lose  a  unit  on  the 
one  and  balancing  it  with  the  loss  or  gain  of  a  unit  on  the  other. 
In  this  way  the  quantities  are  balanced  around  the  average, 
and  the  proportional  parts  of  the  combination  derived.  For 
an  illustration  of  the  method  of  treatment,  see  my  written 
arithmetics. 


CHAPTER  VI. 

HISTORY   OF   PROPORTION. 

THE  Rule  of  Three,  emphatically  called  the  Golden  Rtle, 
by  both  ancient  and  modern  writers  on  arithmetic,  is  found 
in  the  earliest  writings  upon  the  science  of  numbers.  In  the 
Lilawati  the  rule  is  divided,  as  among  modern  writers,  into 
direct  and  inverse,  simple  and  compound,  with  statements  for 
performing  the  requisite  operations,  which  are  said  to  be  quite 
clear  and  definite. 

The  terms  of  the  proportion  in  the  Lilawati  are  written  con- 
secutively, without  any  marks  of  separation  between  them. 
The  first  term  is  called  the  measure  or  argument ;  the  second 
is  its  fruit  or  produce  ;  the  third,  which  is  of  the  same  species 
as  the  first,  is  the  demand,  requisition,  desire,  or  question. 
When  the  fruit  increases  with  the  increase  of  the  requisition, 
as  in  the  direct  rule,  the  second  and  third  terms  must  be  multi 
plied  together  and  divided  by  the  first ;  when  the  fruit  dimin- 
ishes with  the  increase  of  the  requisition,  as  in  the  inverse 
rule,  the  first  and  second  terms  must  be  multiplied  together 
and  divided  by  the  third. 

No  proof  of  the  rule  is  given,  and  no  reference  is  made  to 
the  doctrine  of  proportion  upon  which  it  is  founded.  Under 
compound  proportion  is  given  the  rule  for  five,  seven,  nine  or 
more  terms.  The  terms  in  these  cases  are  divided  into  two 
sets,  the  first  belonging  to  the  argument,  and  the  second  to 
the  requisition ;  the  fruit  in  the  first  set  is  called  the  produce 
of  the  argument ;  that  in  the  second  is  called  the  divisor  of  the 
set ;  they  are  to  be  transposed  or  reciprocally  brought  from  one 
set  to  the  other,  that  is,  the  fruit  is  to  be  put  in  the  second  set 
and  the  divisor  in  the  first. 

(326) 


HISTORY    OF    PROPORTION.  827 

The  Rule  of  Three  Direct  may  be  illustrated  by  the  follow- 
ing example : 

If  two  and  a-half  palas  of  saffron  be  obtained  for  three- 
sevenths  of  a  nishca,  say  instantly,  best  of  merchants,  how 
much  is  got  for  nine  nishcas  ?* 

Statement : 

3         5         9 

7         2         1  Answer,  52  palas  and  2  carshas. 

Rule  of  Three  Inverse  may  be  illustrated  by  the  following 
examples:  If  a  female  slave,  16  years  of  age,  bring  32  nishcas, 
what  will  one  aged  20  cost?  If  an  ox,  which  has  been  worked 
a  second  year,  sell  for  4  nishcas,  what  will  one  which  has  been 
worked  6  years  cost  ? 

1st  question. 

Statement  :  16  32  20.  .  Answer,  25f  nishcas. 

2d  question. 

Statement  :     2     4     6.  Answer,    1£  nishcas. 

In  order  to  understand  the  solution  it  must  be  known  that 
the  value  of  living  beings  was  supposed  to  be  regulated  by  their 
age,  the  maximum  value  of  female  slaves  being  fixed  at  16 
years  of  age,  and  of  oxen  after  2  years'  work;  their  relative 
value  in  the  given  problem  being  as  3  to  1.  The  rule  of  five 
terms  may  be  illustrated  by  the  following  example:  If  the  in- 
terest of  a  hundred  for  a  month  be  five,  what  is  the  interest  of 
sixteen  for  a  year  ? 

Statement : 

1     12,    or  transposing        1     12 
100     16    the  fruit,  100     16 

5  5 

the  product  of  the  larger  set  is  960,  of  the  lesser  100;  the  quo- 
tient is  f  £$  or  ^.,  which  is  the  answer. 

The  interest  of  money,  judging  from  the  examples  in  Brah- 

*To  understand  their  problems  in  rule  of  three  it  must  be  known  that  a 
pala=4  carshas  ;  a  carsha=16  mashas;  and  a  masha=b  gunjas,  or  10  arain*  of 
barley.  Also,  a  nishca=16  drammas;  a  dramma=16 panas ;  apano=4  cacinis, 
and  a  cacini=20  cowry  shells. 


328  THE    PHILOSOPHY    OF    ARITHMETIC. 

megupta  and  Lilawati,  varied  from  3J  to  5  per  cent,  a  month, 
exceeding  greatly  the  enormous  interest  paid  in  ancient  Rome. 
It  is  also  very  high  in  modern  India,  where  it  is  not  uncommon 
for  native  merchants  or  tradesmen  to  give  30  per  cent,  per 
annum. 

The  rule  of  eleven  terms  maybe  illustrated  by  the  following 
example:  Two  elephants  which  are  ten  in  length,  and  nine 
in  breadth,  thirty-six  in  girt,  seven  in  height,  consume  one 
drona  of  grain  ;  how  much  will  be  the  rations  of  statement. 
ten  other  elephants,  which  are  a  quarter  more  in  2     10 

height   and   other   dimensions  ?     The  fruit   and  «     JT 

denominator   being   transposed,   the   answer    is         gg     J 
^52g£.     Dr.  Peacock  remarks  that  the  principle  of  7  '  $£- 

this  very  curious  example  would  be  rather  alarm-  1 

ing,  if  extended  to  other  living  beings  besides  elephants. 

Lucas  di  Borgo  tells  us  that  at  his  time  it  was  usual  for 
students  in  arithmetic  to  commit  to  memory  one  or  other  of 
two  long  rules  which  he  presents.  Tartaglia  mentions  the  first 
of  these  two  rules  in  nearly  the  same  terms  as  Di  Borgo,  and 
gives  also  a  third,  which,  however,  differs  from  it  only  in  ex- 
pression. This  rule  formed  part  of  the  system  in  the  practice 
of  this  subject,  adapted  to  those  who  had  not  sufficient  time  to 
acquire,  genius  to  comprehend,  or  memory  to  retain,  the  rules 
for  the  reduction  and  incorporation  of  fractions;  a  system 
reprobated  by  Tartaglia,  and  attributed  by  him  partly  to  the 
ignorance  of  the  ancient  teachers  of  arithmetic  at  "Venice,  and 
partly  to  the  stinginess  and  avarice  of  their  pupils,  who  grudged 
the  time  and  expense  requisite  for  attaining  a  perfect  under- 
standing of  the  peculiarities  of  fractions. 

An  arithmetician  of  Verona,  named  Francesco  Feliciano  da 
Lazesio,  objects  to  the  memorial  rules  of  Di  Borgo  as  being  too 
general  in  assuming  that  two  of  the  quantities  are  of  one  species, 
and  two,  including  the  term  to  be  found,  of  another  species;  and 
shows  that  in  some  cases  they  are  all  of  the  same  denomina- 
tion.    He   wishes  to   distinguish   the    quantities   into  agents 


HISTORY   OF    PROPORTION.  329 

and  patients,  and  these  again  into  actual,  or  presented  future. 
The  first  term  of  the  proportion  is  the  present  agent,  and  its 
corresponding  patient  is  the  second ;  the  third  term  is  formed 
by  the  future  agent,  and  its  patient  is  the  quantity  to  be  deter- 
mined. This,  it  will  be  noticed,  is  similar  to  the  method  of  cause 
and  effect  adopted  by  some  recent  authors,  and  supposed  to  be 
original  with  them. 

Di  Borgo's  method  of  stating  and  working  a  problem  may  be 
seen  in  the  following  example :  "  If  a  hundred  pounds  of  fine 
sugar  cost  24  ducats,  what  will  be  the  cost  of  975  pounds?" 

via.  va. 

100     24         975 

1 


X 


975 
24 


0 

040 
03400 

3900  23400  (234  ducati. 

1950  10000 

23400  100 

1 
The  following  example  of  the  same  process,  with  fractions  in 
every  term,  is  given  by  Tartaglia :  "  If  3 J  pounds  of  rhubarb 
cost  2J  ducats,  what  will  be  the  cost  of  23 J  pounds  ?" 


lire.         ducati. 

lire. 
95 
4 

\*\\\ 

3 
4 

12 

7 
Partitor  84 

07 

49 

0590 
1330(15  ducati 

844 
8 

95 

7 

665 

2 

000 

1680(20  grossi 

844 
8 

da  partir  1330 
The  quantities,  in  Di  Borgo's  solution,  are  exhibited  under  a 


330  THE    PHILOSOPHY    OF    ARITHMETIC. 

fractional  form,  for  the  purpose  of  making  the  process  more  gen 
eral,  being  equally  applicable  to  fractions  and  whole  numbers. 
It  is  sufficiently  curious  that  he  should  have  considered  it 
necessary  to  construct  the  galea  for  the  division  by  100. 

Different  methods  of  representing  the  terms  of  the  proportion 
were  adopted  by  different  authors.  We  will  state  a  few  of 
them  as  illustrating  the  solution  of  the  problem,  "If  2  apples 
cost  3  soldi,  what  will  13  cost?"  Tartaglia  states  the  propor- 
tion as  follows : 

Se  pomi  2      ||      val  soldi  3      ||      che  valera  pomi  13. 
Other  Italian  authors  write  the  numbers  consecutively  with 
mere  spaces,  and  no  distinctive  marks  between  them ;  thus, 
Pomi.  Soldi.  Pomi. 

2  3  13 

or  thus, 

1  ma.  2  da.  3  tia. 

2  3  13 

In  Recorde  and  older  English  writers,  they  are  written  as 

follows : 

Apples.         Pence. 

2 3 

13  ^^^-^19  J  Answer. 

Humfrey  Baker,  1562,  in  speaking  of  the  rule,  says,  "  The 
rule  of  three  is  the  chiefest,  and  the  most  profitable,  and  most 
excellent  rule  of  all  Arithmetike.  For  all  other  rules  have  neede 
of  it,  and  it  passeth  all  others ;  for  the  which  cause,  it  is  sayde 
the  philosophers  did  name  it  the  Golden  Rule ;  but  now  in  these 
later  days,  it  is  called  by  us  the  Rule  of  Three,  because  it  re- 
quireth  three  numbers  in  the  operation."  He  writes  the  terms 
thus: 

2  3  13 

The  custom  which  generally  prevailed  during  the  lTth  cen- 
tury, was  to  separate  the  numbers  by  a  horizontal  line,  as  fol 
lows: 


HISTORY    OF    PROPORTION. 


331 


Apples. 

2     - 


Pence. 
-     3     ■ 


Apples. 
-     13 


Oughtred,  by  whom  the  subject  of  proportion  was  very  care- 
fully considered,  and  from  whom  the  sign,  : :  ,  to  denote  the 
equality  of  ratios,  seems  to  have  been  derived,  states  a  propor- 
tion as  follows : 

2.  3  : :  13 

In  still  later  times  the  simple  dot  which  separated  the  terms 
of  the  ratios,  was  replaced  by  two  dots,  as  in  the  form  which  is 
now  universally  employed. 

Compound  Proportion,  as  has  been  stated,  was  formerly 
included  under  the  rule  of  five,  six,  etc.,  terms,  there  being  no 
division  of  the  subject  into  simple  and  compound  proportion. 
To  illustrate,  take  the  problem,  "  If  9  porters  drink  in  8  days 
12  casks  of  wine,  how  many  casks  will  serve  24  porters  30 
days  ?"  In  solving  such  problems  Tartaglia  usually  puts  the 
quantity  mentioned  once  only  in  the  last  place  but  one,  instead 
of  in  the  second  place.     The  statement  will  appear  as  follows : 

12     8     30     24 


Divisor,  9x8.     Dividend,  12  X  30  X  24 

Quotient,  -^=120. 

The  example,  "  Twenty  braccia  of  Brescia  are  equal  to  26 

braccia  of  Mantua,  and  28  of  Mantua  to  30  of  Rimini;  what 

number  of  braccia  of  Brescia  corresponds  to  39  of  Rimini?" 

given  by  Tartaglia,  is  solved  as  follows : 

Rimini        Mantua        Mantua        Brescia        Rimini 
30      28      26      20      39 


21840 


Answer,  28. 


332  THE    PHILOSOPHY   OF    ARITHMETIC. 

We  give  another  example  with  its  solution  derived  from  the 
same  author.  "  Six  eggs  are  worth  10  danari,  and  12  danan 
are  worth  4  thrushes,  and  5  thrushes  are  worth  3  quails,  and  8 
quails  are  worth  4  pigeons,  and  9  pigeons  are  worth  2  capons, 
and  6  capons  are  worth  a  staro  of  wheat ;  how  many  eggs  are 

worth  4  star  a  of  wheat  ?" 

960 


1_6_10— 12— 4— 5— 3— 8— 4— 9— 2— 6— 4 

622080  Answer,  648. 

Alligation. — The  rule  for  Medial  Proportion,  or  Alligation, 
is  of  eastern  origin,  and  appears  in  the  Lilawati,  though  under 
a  somewhat  limited  form.  It  is  there  called  suverna-ganita,  or 
computation  of  gold,  and  is  applied  generally  to  the  determin- 
ation of  the  fineness  or  touch  of  the  mass  resulting  from  the 
union  of  different  masses  of  gold  of  different  degrees  of  fine- 
ness. The  questions  mostly  belong  to  what  we  call  Alligation 
Medial.  The  only  question  given  in  illustration  of  Alligation 
Alternate  is  the  following :  "  Two  ingots  of  gold,  of  the  touch  of 
16  and  10  respectively,  being  mixed  together,  the  weight  be- 
came of  the  fineness  of  12 ;  tell  me,  friend,  the  weight  of  gold 
in  both  lumps." 

The  rule  given  for  the  solution  is,  "  Subtract  the  effected  fine- 
ness from  that  of  the  gold  of  a  higher  degree  of  touch,  and  that 
of  the  one  of  the  lower  degree  of  touch  from  the  effected  fine- 
ness ;  tell  me,  friend,  the  weight  of  gold  in  both  lumps  ?  The 
differences  multiplied  by  an  arbitrarily  assumed  number,  will  be 
the  weights  of  gold  of  the  lower  and  higher  degrees  of  purity 
respectively." 

Statement:  16,  10.  Fineness  resulting,  12. 

If  the  assumed  multiplier  be  1,  the  weights  are  2  and  4 
mdshas  respectively ;  if  2,  they  are  4  and  8 ;  if  -J,  they  are  1 
and  2 :  thus  manifold  answers  are  obtained  by  varying  the  as- 
sumption. 


HISTORY   OF   PROPORTION.  383 

This  rule,  though  perfectly  distinct  and  clear,  applies  to  two 
quantities  only,  and  there  is  no  appearance  that  it  was  ever 
applied  to  a  greater  number ;  it  involves,  however,  the  princi- 
ple of  the  rule  which  is  now  used,  recognizes  the  problem  as 
unlimited,  and  shows  in  what  manner  an  indefinite  number  of 
answers  may  be  obtained.  The  extension  of  the  rule  is  not 
entirely  easy,  but  much  more  so  than  the  invention  of  the  orig- 
inal rule  itself;  the  chief  honor  of  the  discovery  of  the  rule 
belongs  therefore  to  the  mathematicians  of  Hindostan.  The 
general  rule  was  known  to  the  Arabians  and  was  denominated 
Sekis,  a  term  meaning  adulterous,  inasmuch  as  it  is  not  con- 
tent with  a  single,  and,  as  it  were,  legitimate  solution  of  the 
question.  It  was  sometimes  called  Cecca  by  the  Italians,  who 
appear  to  have  known  nothing  further  of  the  word  than  its 
Arabic  origin ;  and  it  constitutes  the  alligation  alternate  of 
modern  books  of  arithmetic. 

The  earlier  Italian  writers  on  arithmetic,  in  imitation  of  the 
practice  of  their  Arabian  masters,  have  confined  the  applications 
of  this  rule  almost  entirely  to  questions  connected  with  the  mix 
ture  of  gold,  silver,  and  other  metals,  with  each  other.  This  union 
was  designated  by  the  term  consolare,  which  probably  originated 
in  the  dreams  of  astrologers  and  alchemists,  who  thought  it  the 
peculiar  province  of  the  sun  to  produce  and  generate  gold ;  and 
as  the  process  of  the  alchemist  in  transmuting  the  baser  metals 
into  gold  was  supposed  to  be  under  the  influence  of  the  sun, 
this  gradual  refinement,  which  they  in  common  tended  to  pro- 
duce,  was  designated  by  the  common  term  consolare.  In  later 
times,  it  was  applied  to  silver  as  well  as  gold,  and  still  more 
generally  to  the  common  union  of  these  metals  with  copper. 

To  illustrate  the  method  of  Tartaglia,  take  the  question,  "A 
person  has  five  kinds  of  wheat,  worth  54,  58,  62,  70,  76  lire 
the  staro  respectively ;  what  portion  of  each  must  be  taken,  so 
that  the  sum  may  be  100  stara,  and  the  price  of  the  mixture  66 
lire  the  staro  ?" 


3U 


THE    PHILOSOPHY    OF    ARITHMETIC. 


1st.  In  the  proportion  of  the  numbers  10,  4,  10,  8  and  16. 
54  58  62  70  76 

10  4  10  8  16 


2d.     In  the  proportion  of  the  numbers  14,  14,  14,  24,  24. 
51,  58,  62,  70,  ^6, 

10  10  10  12  12 

4  4  _4  8  8 

"Ti         TT  Ti  _±  _i 

24  24 

Tartaglia  has  given  three  other  solutions  of  this  example  aris- 
ing from  a  different  arrangement  of  the  ligatures.  Among  the 
English  writers  the  method  gradually  assumed  the  form  usually 
found  in  modern  text-books.  The  method  of  explanation  and 
the  extension  of  the  process  as  given  in  a  few  modern  text- 
books may  be  ascribed  to  DeYolson  Wood,  formerly  of  the 
New  York  State  Normal  School. 

Position. — Among  the  most  celebrated  rules  to  which  Pro- 
portion was  applied  in  the  early  text-books  were  those  of  Single 
and  Double  Position.  These  rules  have  been  supplanted  in 
this  country  by  the  simpler  processes  of  arithmetical  analysis, 
but  they  are  still  found  in  English  arithmetics;  and  it  has  been 
suggested  by  a  no  less  eminent  scholar  and  mathematician  than 
Dr.  Hill,  that  they  should  be  retained  in  our  text-books  on 
account  of  their  disciplinary  influences.  Some  historical  facts 
concerning  this  old  rule  will  be  interesting  to  the  reader. 

The  rule  of  Single  Position  is  the  only  one  whijh  is  found 
in  the  Lilawati,  where  it  is  called  Ishtacarman,  or  operation 
with  an  assumed  number.  We  shall  give  a  few  examples  from 
it,  which,  however,  present  nothing  very  remarkable  beyond  the 
peculiarities  of  the  mode  in  which  they  are  expressed. 

1.  Out  of  a  heap  of  pure  lotus  flowers,  a  third  part,  a  fifth. 


HISTORY   OF    PROPORTION.  335 

a  sixth,  were  offered  respectively  to  the  gods  Siva,  Vishnu,  and 
the  Sun,  and  a  quarter  was  presented  to  Bhavani ;  the  remain- 
ing six  were  given  to  the  venerable  preceptor.  Tell  me,  quickly, 
the  whole  number  of  flowers. 

Statement:  £,  hhit  known,  6. 

Put  1  for  the  assumed  number ;  the  sum  of  the  fractions  £, 
i,  J,  J,  subtracted  from  one,  leaves  ^;  divide  6  by  this,  and 
the  result  is  120,  the  number  required. 

2.  Out  of  a  swarm  of  bees,  one-fifth  part  of  them  settled  on 
the  blossom  of  the  cadamba,  and  one-third  on  the  flower  of  a 
silind'hri;  three  times  the  difference  of  these  numbers  flew  to 
the  bloom  of  a  cutaja.  One  bee,  which  remained,  hovered  and 
flew  about  in  the  air,  allured  at  the  same  moment  by  the  pleas- 
ing fragrance  of  a  jasmin  and  pandanus.  Tell  me,  charming 
woman,  the  number  of  bees. 

Statement:  \,  \,  TV  known  quantity,  1;  assumed  30. 

A  fifth  part  of  the  assumed  number  is  6,  a  third  is  10,  differ- 
ence 4  ;  multiplied  by  3  gives  12,  and  the  remainder  is  2.  Then 
the  product  of  the  known  quantity  by  the  assumed  one,  being 
divided  by  the  remainder,  shows  the  number  of  bees  15. 

The  following  question  is  from  the  Manor anjana: 

3.  The  third  part  of  a  necklace  of  pearls,  broken  in  amorous 
struggle,fell  to  the  ground;  its  fifth  part  rested  on  the  couch, 
the  sixth  part  was  saved  by  the  wench,  and  the  tenth  part  was 
taken  up  by  the  lover ;  six  pearls  remained  strung.  Say  of 
how  many  pearls  the  necklace  was  composed. 

Statement:  J,  J,  J,  ^;  remained,  6.  Answer,  30. 

Some  authors  have  attributed  the  invention  of  the  rules  of 
position  to  Diophantus,  though  it  is  impossible  to  discover  upon 
what  grounds.  When  we  consider  the  nature  and  difficulty  of 
the  problems  solved  by  him,  in  those  parts  of  his  works  which 
remain,  we  are  fully  justified  in  supposing  that  the  Greeks  had 
some  method  of  analyzing  and  solving  such  problems,  or  they 
would  not  have  proposed  them  in  such  number  and  variety. 

The  Arabs  were  in  possession  of  the  rules  for  both  Single 


336  THE  PHILOSOPHY    OF    ARITHMETIC. 

and  Double  Position,  with  all  their  applications,  and  in  this 
instance  had  advanced  far  beyond  their  Indian  masters  ;  and 
when  we  consider  how  small  were  the  additions  which  they 
usually  made  to  the  sciences  which  passed  through  their  hands, 
we  might  very  naturally  be  inclined  to  suppose  that  their 
knowledge  of  these  rules  was  derived  from  the  Greeks.  There 
is,  however,  a  vast  gap  in  the  history  of  the  sciences  after 
the  time  of  Theon,  and  it  is  quite  impossible  to  trace  with 
certainty  their  transmission  to  the  Arabs,  or  to  ascertain 
through  what  channels  some  portion  of  Greek  astronomy,  at 
least,  was  transmitted  to  the  Hindoos ;  we  must  therefore  rest 
satisfied  with  the  few  hints  to  be  gathered  from  authors  between 
the  7th  and  12th  centuries,  who  had  access  to  many  writings 
which  have  since  perished. 

The  Italian  writers  on  arithmetic  derived  the  knowledge  of 
these  rules  directly  from  the  Arabians,  distinguishing  them  by 
the  Arabic  name  of  El  Cataym.  The  questions  proposed  by 
Di  Borgo  and  Tartaglia  are  of  immense  variety,  including 
every  case  of  single  and  double  position ;  and  the  rules  which 
are  given  for  this  purpose  are  such  as  would  immediately  result 
from  the  formula  given  in  higher  algebras.  The  following 
example  is  given  and  explained  by  Di  Borgo  : 

4.  A  person  buys  a  jewel  for  a  certain  number  of  fiorini,  I 
know  not  how  many,  and  sells  it  again  for  50.  Upon  making 
his  calculation,  he  finds  that  he  gains  3^  soldi  in  each  fiorino, 
which  contains  100  soldi.     I  ask  what  is  the  prime  cost. 

Suppose  it  to  cost  any  sum  you  choose ;  assume  30  fiorini, 
the  gain  upon  which  will  amount  to  100  soldi,  or  1  fiorino:  1 
added  to  30  makes  31 ;  and  you  say  that  it  makes  50  between 
capital  and  gain ;  the  position  is  therefore  false,  and  the  truth 
will  be  obtained  by  saying,  if  31  in  capital  and  gain  arises  from 
a  mere  capital  of  30,  from  what  sum  will  50  arise.  Multiply 
30  by  50,  the  product  is  1500;  divide  it  by  31,  the  result  is 
4gl|. .  an(j  so  much  I  make  the  prime  cost  of  the  jewel. 

Tartaglia  says  that  such  questions  were  frequently  proposed 


HISTORY   OF   PROPORTION.  337 

as  puzzles  by  way  of  dessert  at  entertainments,  and  has  mixed 
up  with  his  other  questions  a  large  number  of  such  problems. 
The  practice,  from  some  circumstances,  appears  to  be  referable 
to  the  Greek  arithmeticians  of  the  4th  and  5th  centuries,  and 
perhaps  to  an  earlier  period. 

Both  D\  Borgo  and  Tartaglia  sought  to  include  every  possi- 
ble case  of  mercantile  practice  under  the  Rule  of  Three,  giving 
numerous  examples  and  classifying  them  in  various  ways.  The 
Italians  were  also  the  inventors  of  the  rule  of  Practice,  which 
they  regarded  as  an  application  of  the  Rule  of  Three.  Tar- 
taglia gives  some  interesting  and  practical  examples,  with  var- 
ious ingenious  methods  of  solution.  The  great  convenience 
of  these  rules  for  performing  the  calculations  which  were  con- 
tinually occurring  in  trade  and  commerce,  made  them  a  favor- 
ite study  with  practical  arithmeticians,  and  they  assumed  from 
time  to  time  a  constantly  increasing  neatness  and  distinctness 
of  form.  Stevinus,  though,  speaks  of  them  with  some  contempt 
as  forming  "  a  vulgar  compendium  of  the  rule  of  three,  suffi- 
ciently commodious  in  countries  where  they  reckon  by  livres, 
sous  and  deniers."  John  Mellis,  in  his  addition  to  Recorded 
arithmetic,  presents  the  rules  of  Practice  in  a  very  simple  and 
complete  form,  calling  attention  to  them  as  "  briefe  rules  called 
rules  of  practise,  of  rare,  pleasant,  and  commodious  effect, 
abridged  into  a  briefer  method  than  hath  hitherto  been  pub- 
lished." Later  works  gave  them  still  greater  compactness  and 
brevity,  and  in  Cocker's  Arithmetic,  published  in  1G77,  after 
his  death,  and  in  others  printed  towards  the  end  of  the  17th 
century,  they  assumed  the  form  in  which  they  are  now  found 
in  English  arithmetics. 

The  subjects  of  Partnership  and  Barter,  also  treated  by  an 
application  of  Proportion,  seem  to  have  originated  with  the 
Italians.  They  grew  out  of  their  business  transactions,  and  in 
many  cases  were  so  complicated  as  to  require  great  skill  and 
judgment  in  their  solution.  They  are  interesting  as  presenting 
the  type  of  nearly  all  the  questions  of  this  kind  found  in  modern 

text-books. 

22 


SECTION   II. 

THE  PROGRESSIONS. 


I.  Arithmetical  Progression. 


II.  Geometrical  Progression. 


CHAPTER  I. 

ARITHMETICAL   PROGRESSION. 

IN  comparing  numbers  we  perceive  that  we  may  have  a  series 
of  numbers  which  vary  by  a  common  law ;  such  a  series  is 
called  a  Progression.  The  more  general  name  for  such  a  suc- 
cession of  terms  is  Series,  which  is  used  to  embrace  every 
arrangement  of  quantities  that  vary  by  a  common  law,  how- 
ever simple  or  complicated,  and  whether  expressed  in  numbers 
or  in  algebraic  or  transcendental  terms.  The  term  Progression 
is  preferred  in  arithmetic,  and  is  restricted  to  the  arithmetical 
and  geometrical  series. 

The  constant  relation  existing  between  two  or  more  succes- 
sive terms  of  the  series  is  called  the  Law  of  the  progression. 
In  the  series  1,  2,  4,  8,  etc.,  each  term  equals  the  preceding 
term  multiplied  by  2,  and  this  constant  relation  constitutes  the 
law  of  the  series.  It  is  evident  that  the  law  which  connects 
the  terms  of  a  series  may  be  greatly  varied,  and  that  we  may 
thus  have  a  large  number  of  different  kinds  of  series.  The 
only  two  generally  treated  in  arithmetic  are  the  Arithmetical 
and  the  Geometrical  series,  or  progressions. 

Definition. — An  Arithmetical  Progression  is  a  series  of 
terms  which  vary  by  a  constant  difference ;  as  2,  4,  6,  8,  etc. 
The  difference  between  any  two  consecutive  terms  is  called  the 
common  difference.  In  the  series  given,  the  common  difference 
is  2.  The  common  difference  is  sometimes  called  an  arithmet- 
ical ratio  ;  it  is  better,  however,  to  restrict  the  use  of  the  word 
ratio  to  a  geometrical  ratio,  and  call  this  what  it  really  is,  a 
difference. 

(341) 


342  THE    PHILOSOPHY   OF   ARITHMETIC. 

Special  attention  is  called  to  the  definition  of  an  arithmetical 
progression  here  presented.  The  definition  usually  found  in 
our  text-books  is,  "An  arithmetical  progression  is  a  series  of 
numbers  which  increase  or  decrease  by  a  common  difference." 
In  the  definition  proposed  the  word  vary  is  used  to  include 
both  the  increase  and  the  decrease  of  the  terms ;  and  this  is  re- 
garded as  an  improvement  upon  the  old  definition.  It  has 
already  been  adopted  by  two  or  three  authors,  and  should  be 
generally  introduced  into  our  text-books  on  arithmetic. 

Notation. — The  English  and  American  authors  express  an 
arithmetical  progression  by  writing  the  terms  one  after  another 
with  a  comma  between  them.  The  French,  with  more  pre- 
cision, employ  a  special  notation  for  it.  They  place  the  sym- 
bol, -T-,  before  the  progression  and  the  period  (.)  between  the 
terms.  Thus  Bourdon  writes, 

-i-2.  7.  12.  17.  22.  .  .  47.  52.  57.  62. 

This  method  has  been  introduced  into  one  or  two  American 
text-books,  and  may,  in  time,  be  generally  adopted,  though  the 
tendency  seems  to  be  to  adhere  to  the  common  form  of  expres- 
sion. 

Cases. — There  are  five  quantities  in  an  Arithmetical  Progres- 
sion ;  the  first  term,  the  common  difference,  the  number  of 
terms,  the  last  term,  and  the  sum  of  all  the  terms.  If  any  three 
of  these  are  given,  the  other  two  can  be  found  from  them. 
This  gives  rise  to  twenty  different  cases,  in  which  any  three 
terms  being  given,  the  other  two  may  be  found.  These  cases 
cannot  all  be  solved  by  arithmetic,  since  some  of  them  involve 
the  solution  of  a  quadratic  equation  ;  they*  are,  however,  very 
readily  treated  by  the  principles  of  algebra.  The  two  principal 
cases  in  arithmetic  are  as  follows: 

1.  To  find  the  last  term,  having  given  the  first  term,  the 
common  difference,  and  the  number  of  terms. 

2.  To  find  the  sum  of  the  terms,  having  given  the  first  term, 
the  last  term,  and  the  number  of  terms. 

Method  of  Treatment. — The  treatment  of  Arithmetical  Pro- 


ARITHMETICAL   PROGRESSION.  34$ 

gression  in  arithmetic  is  very  simple.  We  derive  the  rule  for 
finding  the  last  term  by  noticing  the  law  of  the  formation  of  a 
few  terms  and  then  generalizing  this  law.  Thus  we  notice  that 
the  second  term  of  an  arithmetical  progression  equals  the  first 
term  plus  once  the  common  difference,  the  third  term  equals 
the  first  term  plus  twice  the  common  difference,  etc. ;  hence  we 
infer  that  the  last  term  equals  the  first  term  plus  the  product 
of  the  common  difference  by  the  number  of  terms  less  one. 

In  finding  the  sum  of  the  terms  we  take  a  series,  then  write 
under  this  series  the  same  series  in  an  inverted  order,  then 
adding  the  two  series  we  see  that  twice  the  sum  of  the  series 
is  the  same  as  the  sum  of  the  extremes  multiplied  by  the  num- 
ber of  terms ;  and  generalizing  this  we  obtain  the  rule  for  find- 
ing the  sum. 

In  algebra  we  reason  in  the  same  way,  except  that  we  employ 
general  symbols,  and  use  a  general  series  instead  of  a  special 
one.  Expressing  the  two  fundamental  rules  in  general  formu- 
lae, we  can  readily  find  the  rest  of  the  twenty  cases  by  the  alge- 
braic process  of  reasoning.  These  two  simple  cases,  I  think, 
should  in  arithmetic  be  expressed  in  the  concise  language  of 
algebraic  symbols.  Pupils  who  have  not  studied  algebra  will 
have  no  difficulty  in  understanding  them.  The  two  rules  of 
arithmetical  progression  are  briefly  expressed  thus: 

1. 1= a  +  (n— 1) .  d;  2.  s  =  (a  +  I) .  5 

History. Of  the  origin  of  the  progressions  and  the  methods 

of  treatment,  but  little  is  known.  They  were  the  object  of  the 
particular  attention  of  the  Fythagorean  and  Platonic  arithme- 
ticians, who  enlarged  upon  the  most  trivial  properties  of  num- 
bers with  the  most  tedious  minuteness.  Directing  their  spec- 
ulations, however,  to  the  mysterious  harmonies  of  the  physical 
and  intellectual  world,  they  passed  over,  as  unworthy  of  no- 
tice, the  solutions  of  those  problems  which  naturally  arise  from 
these  progressions,  and  which  appear  in  such  numbers  in  Ilin 
doo,  Arabic,  and  modern  European  books  on  Arithmetic. 


344  THE   PHILOSOPHY   OF   ARITHMETIC. 

Very  little  is  known  concerning  the  origin  of  the  familiar 
problems  usually  found  under  this  subject.  The  problem, 
"  How  many  strokes  do  the  clocks  in  Venice  strike  in  24 
hours V1  is  supposed  to  be  of  Venetian  origin.  The  following 
familiar  problem  is  attributed  to  Bede :  "There  is  a  ladder 
with  100  steps;  on  the  first  step  is  seated  one  pigeon,  on  the 
second  step  two  pigeons,  on  the  third  step  three,  and  so  on 
increasing  by  one  each  step ;  tell,  who  can,  how  many  pigeons 
were  placed  on  the  ladder."  The  celebrated  problem, — "If 
a  hundred  stones  be  placed  in  a  right  line,  one  yard  apart 
and  the  first  one  yard  from  a  basket,  what  length  of  ground 
must  a  person  go  over  who  gathers  them  up  singly,  returning 
with  them  one  by  one  to  the  basket  ?" — though  found  in  many 
modern  text-books,  is  very  old,  but  its  origin  is  not  known. 

The  extraordinary  magnitude  of  the  numbers  which  result 
from  the  summation  of  a  geometrical  series  is  well  calculated 
to  excite  the  surprise  and  admiration  of  persons  who  are  not 
fully  aware  of  the  principle  upon  which  the  increase  of  the 
terms  depends;  and  examples  are  not  wanting  among  the 
earliest  writers,  where  the  rash  and  ignorant  are  represented 
as  being  seduced  into  ruinous  or  impossible  engagements. 
The  most  celebrated  of  these  is  that  which  tradition  has 
represented  as  the  terms  of  the  reward  demanded  of  an  Indian 
prince  by  the  inventor  of  the  game  of  chess  ;  which  was  a 
grain  of  wheat  for  the  first  square  on  the  chess  board,  two 
grains  for  the  second  square,  four  for  the  third,  and  so  on, 
doubling  continually  to  sixty-four,  the  whole  number  of 
squares. 

Lucas  di  Borgo  solved  the  question,  and  found  the  result 
to  be  18446744073709551615,  which  he  reduces  to  higher 
denominations  and  finds  it  equal  to  209022  castles  of  corn. 
He  then  recommends  his  readers  to  attend  to  this  result,  as 
they  would  then  have  a  ready  answer  to  many  of  those 
barbioni  ignari  de  la  arithmetica  who  have  made  wagers  on 
such  questions,  and  have  lost  their  money. 


CHAPTER  II. 

GEOMETRICAL   PROGRESSION. 

A  GEOMETRICAL  PROGRESSION  is  a  series  of  terms 
which  vary  by  a  common  multiplier  ;  as,  1,  2,  4,  8,  16,  etc. 
The  common  multiplier  is  called  the  rale  or  ratio  of  the  pro- 
gression ;  thus,  in  the  progression  given,  the  rate  is  2.  The 
rate  of  the  progression  equals  the  ratio  of  any  term  to  the  pre- 
ceding term.  When  the  progression  is  ascending,  the  rate  is 
greater  than  a  unit;  when  it  is  descending,  the  rate  is  less 
than  a  unit.  The  rate  is  by  most  authors  called  the  ratio  of 
the  series;  the  reason  for  preferring  the  term  rate  will  be 
stated  presently. 

Notation. — The  method  of  writing  a  geometrical  progression, 
generally  employed  by  English  and  American  authors,  is  the 
same  as  that  for  an  arithmetical  progression.  The  French 
authors,  however,  distinguish  it  from  an  arithmetical  progression 
by  a  special  notation.  They  place  the  symbol  -h-  before  the 
series,  and  separate  the  terms  by  a  colon  (:)  ;  thus, 
-rr  2  :  4  :  8  :  16  :  32  :  64  :  128. 

Tlie  Rate. — The  constant  multipler,  as  before  stated,  is  gen- 
erally called  the  ratio  of  the  series.  The  term  rate,  it  is 
thought,  is  much  more  appropriate  and  precise.  The  objection 
to  the  word  ratio  is  that,  in  the  comparison  of  numbers,  the 
ratio  is  the  quotient  of  the  first  term  divided  by  the  second, 
while  the  rate  of  a  series  is  equal  to  any  term  divided  by  the 
previous  term ;  hence,  there  is  a  seeming  contradiction  of  the 
correct  meaning  of  the  term  ratio.  This  contradiction  may  be 
only  seeming,  but  to  avoid  all  difficulty  in  this  respect,  it  will 
15*  (345) 


346  THE   PHILOSOPHY    OF    ARITHMETIC. 

be  better  to  use  a  term  which  is  appropriate  and  not  liable  to 
misconception.  Rate  seems  to  be  an  appropriate  word,  since 
we  naturally  speak  of  the  rate  of  increase  or  decrease  of  any- 
thing; and  by  the  rate  of  a  progression,  we  mean  its  rate  of 
increase  or  decrease. 

The  French  mathematicians  make  this  distinction  between 
ratio  and  rate  ;  they  use  the  word  rapport,  ratio,  in  proportion, 
and  raison,  rate,  in  progression.  Bourdon  says,  "The  con- 
stant ratio,  which  exists  between  any  term  and  that  which  imme* 
diately  precedes  it,  is  called  the  rale  of  the  progression*  By 
rapport  they  seem  to  mean  about  what  we  do  by  ratio  ;  it  is 
probably  from  the  idea  of  produce,  the  ratio  being  the  product 
of  the  division.  Their  word  raison  seems  to  mean  the  same  as 
rate,  taken  probably  from  the  idea  of  cause,  the  rate  being  the 
law  or  cause  of  the  terms  being  what  they  are. 

The  term  ratio,  as  used  in  relation  to  a  progression,  has 
given  rise  to  a  good  deal  of  discussion  and  misapprehension. 
Some  writers  who  use  the  word  have  taken  the  pains  to  tell  us 
that  they  mean,  not  a  direct,  but  an  inverse  ratio.  Prof.  Dodd 
says,  when  we  speak  of  the  ratio  of  a  geometrical  progression 
being  2,  we  mean  that  "  the  terms  progress  in  a  twofold  ratio, 
which  simply  means  that  each  term  has  the  ratio  of  2  to  the 
preceding  term ;"  and  similar  remarks  are  made  by  other  writers. 
By  using  the  word  rate  instead  of  ratio,  all  this  difficulty  and 
misapprehension  will  be  avoided.  It  is  to  be  hoped,  therefore, 
that  the  term  rate  will  be  generally  adopted  in  speaking  of  the 
law  of  variation  of  a  geometrical  series. 

Cases.— There  are  five  quantities  considered,  as  in  arithmet- 
ical progression  ;  the  first  term,  the  rate,  the  number  of  terms, 
the  last  term,  and  the  sum  of  the  terms.  Any  three  of  these 
being  given  the  other  two  can  be  derived  from  them,  which 
gives  rise  to  twenty  distinct  cases.     These  cannot  all  be  solved 

*Ce  ra,,port  constant,  qni  existe  entre  un  tcnne  et  celui  qui  le  proce«l« 
immetliatoment,  sc  nouuuc  la  Uaison  do  la  progression.— Uouudon's  Arith, 
rustic,  pagu  '279. 


GEOMETRICAL   PROGRESSION.  347 

by  arithmetic ;  the  first  fifteen  are  easily  derived  by  common 
algebra,  and  the  other  five  readily  yield  to  the  logarithmic  cal- 
culus. The  two  cases  generally  given  in  arithmetic  are  the 
following: 

1.  To  find  the  last  term,  having  given  the  first  term,  the  rate, 
and  the  number  of  terms. 

2.  To  find  the  sum  of  the  terms,  having  given  the  first  term, 
the  last  term,  and  the  number  of  terms. 

Treatment. — The  general  method  of  treatment  in  a  geomet- 
rical progression  is  the  same  as  in  an  arithmetical  progression ; 
and  having  been  stated  under  arithmetical  progression,  need 
not  be  repeated  here.  Several  cases  cannot  be  obtained  in 
arithmetic,  since  they  require  the  solution  of  an  equation.  Four 
cases  cannot  be  solved  by  elementary  algebra,  as  they  depend 
upon  the  solution  of  an  exponential  equation  ;  and  in  obtaining 
the  numerical  results  we  are  obliged  to  make  use  of  logarithms. 
The  two  fundamental  cases  should,  we  think,  in  arithmetic  be 
expressed  in  the  symbolic  language  of  algebra;  thus, — 

1.  l=arn-*:  2.   £=^=^. 

r — 1 

TnE  Infinite  Series. — An  Infinite  Series  is  a  series  in 
which  the  number  of  terms  is  infinite.  In  a  descending  pro- 
gression the  terms  are  continually  growing  smaller;  hence  if 
the  series  be  continued  sufficiently  far,  the  last  term  must  be- 
come less  than  any  assignable  quantity;  and  if  continued  to 
infinity,  the  last  term  must  become  infinitely  small. 

In  treating  an  infinite  scries,  we  regard  this  infinitely  small 
quantity  as  zero,  or  nothing.     Thus,  in  finding  the  sum  of  a 

descending  series,  we  use  the  formula  S=- ;  and  regarding 

1 — r 

the  last  term  as  nothing,  the  term  Ir  disappears,  and  we  have 

S=z ,  or  the  sum  of  the  terms  of  an  infinite  series  descend- 

1 — r 

ing  equals  the  first  term  divided  by  1  minus  the  rate. 


348  THE    PHILOSOPHY    OF    ARITHMETIC. 

This  reduction  of  the  last  term  to  zero  presents  a  difficulty 
not  easily  explained.  The  question  arises,  how  can  the  last 
term  become  zero?  At  what  point  does  a  term  become  so 
small  that,  when  multiplied  by  the  rate,  the  product  shall  be 
nothing?  To  illustrate  the  difficulty,  take  the  series  1,  J,  £,  J, 
etc.,  in  which  the  rate  is  \.  Now  if  this  series  be  continued  to 
infinity,  the  last  term  is  supposed  to  be  zero.  This  supposition 
seems  to  involve  the  idea  that  the  term  just  before  the  last  is 
so  small  that  J  of  it  is  nothing.  Who  can  conceive  of  such  a 
term?  Who  can  trace  the  series  down  through  all  the  differ- 
ent values,  until  we  reach  a  term  so  small  that  one-half  of  it  is 
nothing?  This  of  course  cannot  be  done.  The  mind  shrinks 
from  the  effort;  it  is  unable  to  grasp  the  infinitely  small.  In- 
deed, neither  the  infinitely  great  nor  the  infinitely  small  can  be 
positively  conceived ;  an  infinite  quantity  and  an  infinitesimal 
are  both  beyond  the  grasp  of  the  human  mind. 

What  shall  we  do  then  ?  Shall  we  deny  that  the  last  term 
is  infinitely  small,  or  zero?  Certainly  not:  to  assume  that  it 
is  not  infinitely  small  involves  a  greater  difficulty  than  the  sup- 
position that  it  is  infinitely  small.  Fix  upon  any  term,  how- 
ever small,  and  we  see  that  it  can  be  continually  divided,  and 
that  the  division  will  continue  as  long  as  there  is  a  term  to  be 
divided,  and  can  only  terminate  when  the  term  becomes  too 
small  to  divide,  or  zero.  Hence,  to  conceive  that  the  infinite 
term  is  not  zero,  is  to  suppose  that  the  division  stopped  when 
it  could  have  proceeded,  which  is  absurd ;  consequently,  it  is 
absurd  to  suppose  that  the  last  term  is  not  zero.  The  question 
then  stands  thus:  we  cannot  comprehend  that  the  last  term  is 
zero,  and  to  conceive  that  it  is  not  zero  is  absurd.  We  are 
thus  in  the  dilemma  that  we  must  believe  either  the  absurd  or 
the  incomprehensible.  We  cannot  believe  the  absurd;  we 
rather  accept  the  incomprehensible.  We  are  therefore  forced 
to  the  conviction  that  the  last  term  is  zero,  even  though  we 
cannot  fully  conceive  it  to  be  so.  We  believe  that  which  we 
cannot  fully  understand,  because  not  to  believe  it  leads  to  an  ab 


GEOMETRICAL    PROGRESSION.  349 

\irdity,  and  the  mind  is  so  constituted  that  it  will  accept  the 
ncoraprehensible  sooner  than  the  absurd.  We  take  it  upon 
faith;  it  is  the  place  in  science  "where  reason  falters "  and 
faith  accepts. 

This  method  of  considering  the  subject  presents  an  excellent 
illustration  of  the  operation  of  the  intuitive  power  in  many 
questions  of  religious  faith.  I  may  not  be  able  to  comprehend 
a  first  cause ;  but  1  know  there  must  be  one,  or  else  I  am  in- 
volved in  an  absurdity,  and  the  human  mind  cannot  rest  in  the 
absurd.  It  may  be  remarked  that  the  point  of  difficulty  here' 
considered,  is  one  that  frequently  occurs  in  mathematics.  The 
infinitely  small  is  an  important  element  in  mathematical  inves- 
tigations. We  make  use  of  it  in  geometry,  and  in  calculus  it  is 
the  fundamental  idea  upon  which  the  science  is  based. 

The  most  satisfactory  method  of  removing  any  doubt  that 
one  may  have  upon  the  assumption  that  the  last  term  reduces 
to  zero,  is  to  take  a  problem  which  may  be  solved  by  an  infinite 
series,  and  which  can  also  be  solved  without  it.  If  the  result 
obtained  by  supposing  the  last  term  to  be  zero,  agrees  with  the 
result  otherwise  obtained,  the  conclusion  that  the  last  term 
is  zero  must  be  accepted,  whether  we  can  conceive  it  or  not. 
Such  a  problem  is  the  following:  "A  hound  and  fox  are  10 
rods  apart,  and  the  hound  pursues  the  fox ;  how  far  will  the 
hound  run  to  overtake  the  fox,  if  the  latter  runs  y1^  as  fast  as 
the  hound?" 

Looking  at  this  problem  in  one  way,  we  see  that  when  the 
hound  has  run  the  10  rods  the  fox  has  run  1  rod,  and  they 
are  then  1  rod  apart.  When  the  hound  runs  this  rod,  the 
fox  has  run  ^  of  a  rod ;  hence  they  are  then  -^  of  a  rod  apart. 
When  the  hound  runs  this  y1^  of  a  rod,  they  are  y1^  of  y1^,  or  T^ 
of  a  rod  apart;  hence  the  distance  the  hound  will  run  to  catch 
the  fox  is  correctly  represented  by  the  sum  of  the  series  10-f  1 
+ A+  r&y+TT)W+ 1  o i  o  o  +  etc,  to  an  infinite  number  of  terms 
The  sum  of  this  series,  obtained  by  the  method  of  infinite  series, 
which  regards  the  last  term  as  zero,  equals  10-i-(l — rff)= 10-^1% 


350  THE   PHILOSOPHY   OV   ARITHMETIC. 

__10x^=ll£rods.     Hence  the  hound  runs  11J  rods  to  catch 
the  fox. 

The  problem  may  also  be  solved  by  the  following  simplo 
method  of  analysis :  By  the  conditions,  ten  times  the  distance 
the  fox  runs  equals  the  distance  the  hound  runs ;  and  this  di- 
minished by  the  distance  the  fox  runs,  is  9  times  the  distance 
the  fox  runs,  which  equals  what  the  hound  gains  on  the  fox, 
or  10  rods,  the  distance  they  were  apart;  then  once  the  dis- 
tance the  fox  runs  equals  ^  of  a  rod,  and  10  times  the  distance 
the  fox  runs,  which  is  the  distance  the  hound  runs,  equals 
10xi^-=-L^,  or  11£  rods.  Or,  we  may  solve  it  even  more 
simply  thus:  the  hound  gains  9  rods  in  running  10,  hence  to 
gain  1  rod  he  will  run  -^  of  a  rod,  and  to  gain  10  rods,  so  as 
to  catch  the  fox,  he  will  run  10  times  ^,  or  ifQ-^lli  rods. 
This  result  corresponds  with  that  obtained  by  the  summation 
of  the  infinite  series;  hence  the  supposition  involved  in  that 
solution,  that  the  last  term  of  the  series  equals  zero,  must  be 
correct. 

This  problem  is  sometimes  given  as  a  puzzle,  in  which  it  is 
said  that  since  there  is  always  one-tenth  of  the  previous  dis- 
tance between  them,  the  hound  will  never  catch  the  fox.  The 
fallacy  consists  in  inferring  that  because  there  is  an  infinite 
number  of  successive  operations,  it  must  require  an  infinite 
length  of  time  to  perform  them. 

A  problem  similar  to  this  is  the  following :  "A  ball  falls  8  feet 
to  the  floor  and  bounds  back  4  feet,  then  falling  bounds  2  feet, 
and  so  on;  how  far  will  it  move  before  coming  to  rest?"  Solv- 
ing this,  we  find  the  distance  to  be  24  feet.  It  is  sometimes 
supposed  in  this  problem,  that  the  body  will  never  come  to 
rest ;  this  is  a  mistake,  for  though  there  will  be,  in  theory  at 
least,  an  infinite  number  of  motions,  they  will  be  accomplished 
in  a  finite  period  of  time.  The  reason  of  this  is,  that  the  infi- 
nitely small  motions  are  made  in  infinitely  small  periods  of 
time,  the  sum  of  which  does  not  exceed  a  finite  period. 

It  should  be  remarked  that  some  writers  maintain  that  the 


GEOMETRICAL   PROGRESSION.  351 

results  in  the  infinite  series  are  not  absolutely  correct,  but  are 
merely  approximations ;  thus,  that  the  sum  of  the  series  i+j+g- 
-f-etc,  is  not  absolutely  1,  but  only  approximately  so;  in  other 
words,  that  all  we  can  affirm  concerning  it  is  that  it  comes  nearer 
and  nearer  to  1  as  we  increase  the  number  of  terms,  though  it  can 
never  reach  1.  Unity  is  the  limit  towards  which  it  is  always 
approaching,  which  it  never  can  exceed,  and  indeed,  which  it 
never  can  reach. 

This  is  the  doctrine  of  limits,  and  is  the  one  usually  preferred 
by  modern  mathematicians.  By  this  doctrine,  in  summing  the 
infinite  series  descending,  we  are  attempting  to  find  the  limit 
towards  which  the  series  is  approaching,  but  which  it  can  never 
reach.  This  is  regarded  as  the  most  logical  method  of  consider- 
ing the  subject.  Logic  may  admit  self-evident  propositions,  but 
it  does  not  admit  conclusions  that  cannot  be  logically  derived 
from  these  self-evident  assumptions.  Thought  cannot  follow  an 
infinite  series  step  by  step  to  the  zero  term  ;  hence  a  conclusion 
based  on  the  assumption  of  a  zero  term  is  regarded  as  illogical 
and  inadmissible. 

This  doctrine  of  limits  as  applied  to  the  infinite  series,  while 
apparently  logical,  is  not  without  its  difficulties.  It  would  seem 
to  lead  to  the  conclusion  that  in  the  case  of  the  "  fox  and  hound 
problem,"  given  above,  the  hound  would  never  catch  the  fox ; 
unless,  as  a  boy  once  remarked,  "  he  gets  near  enough  to  grab 
him."  So  in  respect  to  the  elastic  ball  dropped  upon  a  pave- 
ment; if  the  result  is  only  approximately  true,  does  it  not  follow 
that  the  ball  never  comes  to  rest,  but  continues  bounding  forever  ? 
Here,  as  in  many  other  cases,  faith  in  the  incomprehensible  seems 
more  satisfactory  than  a  timid  skepticism. 

It  will  be  interesting  to  notice  that  the  two  different  series,  J+ 

i"'~T7'~^Tr"'~etc'>  an^  i+i+iV^A""^610**  are  eacn  eoiual  to  tne 
same  fraction  J.  It  is  also  an  interesting  truth  that  the  sum  of 
the  series  beginning  with  J,  and  decreasing  at  the  rate  of  -J,  is 
just  equal  to  1. 


SECTION    III 

PERCENTAGE. 


23 


I.    Nature  of  Percentage. 


II.     Nature  of  Interest. 


CHAPTER  I. 

NATURE   OF   PERCENTAGE. 

PERCENTAGE  is  a  process  of  computation  in  which  the 
basis  of  the  comparison  of  numbers  is  a  hundred.  The 
same  idea  may  also  be  expressed  more  briefly  in  the  definition, 
Percentage  is  the  process  of  computing  in  hundredths. 

The  former  definition  was  first  presented  in  one  of  the  author's 
arithmetical  works.  Up  to  this  time  no  definition  had  been 
given  of  Percentage  as  a  process  of  arithmetic.  In  the  text- 
books, the  word  was  merely  defined  as  meaning  so  many  of  a 
hundred.  Soon  after  this  publication  appeared,  one  or  two 
other  authors  adopted  a  definition  similar  to  the  one  given 
above,  presenting  the  subject  as  a  department  of  the  science ; 
and  in  time,  it  is  presumed,  all  will  define  it  as  a  process  of 
arithmetic. 

It  will  be  readily  seen  that  Percentage  has  its  origin  in  the 
third  division  of  the  science  of  arithmetic;  namely,  Comparison. 
We  may  compare  numbers  and  determine  their  relations  with 
respect  to  their  common  unit  or  basis.  This  is  the  first  and 
simplest  case  of  comparison,  and  gives  rise  to  Ratio  and  Pro- 
portion. We  may  also  compare  numbers  with  respect  to  some 
number  agreed  upon  as  a  basis  of  comparison,  and  develop 
their  relations  with  respect  to  this  basis.  When  this  number 
is  one  hundred,  we  have  the  process  of  Percentage.  It  is  thus 
seen  that  the  idea  of  the  subject  presented  in  the  definition 
given  above  is  correct. 

Percentage  originated  in  the  fact  of  the  convenience  of  esti- 
mating by  the  hundred,  in  a  decimal  scale.     It  derives  its  im- 

(355) 


356  THE   PHILOSOPHY   OF    ARITHMETIC. 

portance  and  has  received  so  full  a  development,  partly  at  least, 
from  the  fact  of  our  having  a  decimal  currency.  It  occupies  a 
more  prominent  place  in  American  than  in  English  text-books, 
where  the  money  system  is  not  decimal.  Its  principal  use  is  in 
its  application  to  business  transactions  relating  to  money,  as 
will  be  seen  in  the  various  ways  in  which  it  is  employed.  It 
admits,  however,  of  a  purely  abstract  development,  entirely 
independent  of  concrete  examples ;  and  is,  therefore,  a  process 
of  pure  arithmetic. 

Quantities. — Percentage  embraces  four  distinct  kinds  of  quan- 
tities, the  base,  the  rate,  the  percentage,  and  the  amount  or 
difference. 

The  Base  is  the  number  on  which  the  percentage  is  estimated. 
The  Rate  is  the  number  of  hundredths  of  the  base.  The  Per- 
centage  is  the  result  of  taking  a  number  of  hundredths  of  the 
base.  The  Amount  or  Difference  is  the  sum  or  the  difference 
of  the  base  and  percentage. 

The  Amount  and  Difference  are  the  same  kind  of  quantities, 
and  it  would  be  well,  in  Percentage,  to  have  some  one  term 
which  would  include  them  both.  In  several  of  the  applications 
we  have  such  a  word ;  as  selling  price  in  Profit  and  Loss, 
proceeds  in  Discount,  etc.  The  expression  Resulting  Number 
has  been  used,  but  this  is  a  little  awkward  and  inconvenient. 
The  term  Proceeds,  meaning  that  which  results  or  comes  forth, 
I  have  sometimes  thought  of  adopting,  and  indeed  have  adopted 
in  one  of  my  works.  Some  term,  in  place  of  amount  and 
difference  as  used  in  percentage,  is  a  scientific  necessity,  and 
Proceeds  is  recommended. 

The  Rate  was  originally  expressed  as  a  whole  number,  and 
the  methods  of  operation  based  upon  such  expression.  Latterly 
it  is  becoming  the  custom  to  represent  the  rate  as  a  decimal, 
and  to  operate  with  it  as  such.  This  is  much  the  better  way, 
and  will  probably  become  universal.  It  gives  greater  simplicity 
to  the  rules,  makes  the  treatment  more  scientific,  and  is  quite 
as  readily  understood  by  pupils.      It  may  be  remarked  that 


NATURE    OF    PERCENTAGE.  357 

the  definition  of  the  rate  will  vary  according  to  which  of  these 
forms  is  taken.  The  definition  above  given  regards  the  rate  as  a 
decimal. 

It  will  thus  appear  that  there  is  a  slight  distinction  between 
the  term  Rate  and  the  expression  the  rate  per  cent.  Per  cent. 
means  by  the  hundred ;  rate  per  cent,  means  a  certain  number 
of  or  by  the  hundred ;  while  Rate  means  a  certain  number  of 
hundredths.  When  money  is  loaned  at  6  per  cent,  the  rate 
per  cent,  is  6;  but  the  Rate  is  .06.  Thus  Rate  and  rate  by  the 
hundred,  are  about  identical  in  meaning.  We  may  conse- 
quently define  the  Rate  to  be  the  number  by  which  we  multiply 
the  base  in  order  to  obtain  any  required  per  cent,  of  it;  and 
this  is  what  is  intended  in  the  definition, — The  rate  is  a  num- 
ber of  hundredths  of  the  base. 

Gases. — It  has  been  a  question  among  arithmeticians  under 
how  many  cases  Percentage  should  be  presented.  There  being 
four  distinct  classes  of  quantities — five,  if  like  some  authors  we 
regard  the  amount  and  difference  as  distinct — any  two  of  which 
being  given,  the  others  may  be  found,  it  will  be  seen  that  there 
are  quite  a  large  number  of  possible  theoretical  cases.  What  is 
the  simplest  and  most  scientific  classification  of  these  various 
cases?  In  other  words,  what  are  the  general  cases  of  Per- 
centage? It  has  been  quite  customary  to  present  the  subject 
under  six  distinct  cases,  and  this  affords  a  very  practical  view 
of  the  subject.  Authors,  however,  have  not  been  uniform  in 
their  treatment.  I  believe  that  the  best  way  is  to  present  the 
subject  under  three  general  cases,  each  of  which  will  contain 
two  or  three  special  cases,  as  we  regard  the  amount  and  differ- 
ence as  one  or  two  classes  of  quantities.  Uniting  the  amount 
and  difference  under  one  general  term,  as  proceeds,  we  shall 
have  three  general  cases,  each  including  two  special  cases, 
making  six  cases  in  all ;  regarding  the  amount  and  difference 
as  two  distinct  quantities,  we  shall  have  three  special  cases 
under  each  general  case,  making  nine  cases  in  all. 

These  three  general  cases  may  be  formally  stated  as  follows : 


358  THE   PHILOSOPHY   OF   ARITHMETIC. 

1.  Given,  the  base  and  the  rate,  to  find  the  percentage  and 
the  proceeds. 

2.  Given,  the  base  and  either  the  percentage  or  the  proceeds, 
to  find  the  rate. 

3.  Given,  the  rate  and  either  the  percentage  or  the  proceeds, 
to  find  the  base. 

Treatment. — There  are  two  distinct  methods  of  treatment  in 
Percentage,  which  may  be  distinguished  as  the  Analytic  and  the 
Synthetic  methods.  The  Analytic  Method  consists  in  reducing 
the  rate  to  a  common  fraction,  and  taking  a  fractional  part  of 
the  base  for  the  percentage,  and  operating  similarly  in  the  other 
cases.  It  differs  particularly  from  the  other  method  in  the 
solution  of  the  second  and  third  cases,  as  will  be  seen  by  the 
solution  of  a  problem.  It  is  the  method  for  mental  analysis, 
and  is  especially  suited  to  the  subject  of  Mental  Arithmetic. 
To  illustrate  the  analytic  method,  take  the  problem,  "  What  is 
25%  of  360?"  We  reason  thus:  25%  of  360  is  T2^  or  \  of 
360,  which  is  90.  To  find  the  base  take  the  problem, — "  90  is 
25%  of  what  number?"  The  solution  is,— If  90  is  25%,  or 
\,  of  some  number,  £  of  the  number  is  4  times  90,  or  360.  The 
case  of  finding  the  rate  per  cent,  is  solved  in  a  similar  manner. 

The  Synthetic  Method  consists  in  preserving  the  rate  in  the 
form  in  which  it  is  presented,  and  operating  accordingly.  In 
the  synthetic  method  there  are  two  ways  of  operating :  the 
first  consists  in  using  the  rate  as  a  whole  number,  and  dividing 
or  multiplying  by  a  hundred ;  the  second  operates  with  the 
rate  in  the  form  of  a  decimal,  according  to  the  principles  of 
decimal  multiplication  and  division.  There  has,  for  several 
years,  been  a  tendency  towards  the  latter  method,  and  arithme- 
ticians are  now  generally  agreed  in  its  favor. 

This  latter  method  is  greatly  to  be  preferred  on  account  of 
its  simplicity  and  scientific  character.  The  difference  may 
be  shown  by  a  rule  for  one  of  the  cases.  When  the  rate  is 
used  as  a  whole  number,  the  rule  for  finding  the  percentage  is, 
—Multiply  the  base  by  the  rate,  and  divide  the  product  by  100. 


NATURE  OF   PERCENTAGE.  359 

When  the  rate  is  used  as  a  decimal,  the  rule  is, —  Multiply  the 
base  by  the  rate.  A  similar  difference  will  be  found  to  exist  in 
the  rules  for  all  the  cases.  Another  consideration  in  favor  of 
using  the  rate  as  a  decimal  is  the  ease  with  which  the  rules  for 
the  other  cases  are  derived  from  the  first.  Assuming  that  the 
percentage  equals  the  base  multiplied  by  the  rate ;  it  immedi- 
ately follows  that  the  base  equals  the  percentage  divided  by 
the  rate,  or  the  rate  equals  the  percentage  divided  by  the  base. 

To  illustrate  the  method  preferred,  suppose  we  have  the 
problem  in  Case  1.,— "What  is  25%  of  360?"  We  would  rea- 
son thus :  Twenty-five  per  cent,  of  360  equals  25  hundredths 
times  360,  or  360 x. 25,  which  by  multiplying  we  find  to  be  90. 

To  illustrate  Case  2,  take  the  problem,  "  90  is  25%  of  what 
number?"  We  would  solve  this  as  follows:  If  90  is  25%  of 
some  number,  then  some  number  multiplied  by  .25  equals  90; 
hence  this  number  equals  90  divided  by  .25,  or  90-7-.25,  which 
by  dividing  we  find  is  360. 

To  illustrate  Case  3,  take  the  problem, — "  90  is  what 
per  cent,  of  360  ?"  The  solution  is  as  follows  :  If  90  is  some 
per  cent,  of  360,  then  360  multiplied  by  some  rate  equals  90 ; 
hence  the  rate  equals  90  divided  by  360,  or  90-i-360,  which  is 
.25,  or  25%. 

The  solution  of  problems  including  the  proceeds  is  quite 
similar,  and  need  not  be  presented  here  in  detail.  The  particu- 
lar method  of  explanation  will  be  found  in  my  Higher  Arith- 
metic. 

Formulas. — These  synthetic  methods  and  rules  may  all  be 
presented  in  general  formulas,  as  follows: 

Case  I.  Case  II.  Case  III. 

1.  bxr—p  1.  p-=-r=6  1.  p-±-b=r 

2.  bx(\+r)=A       2.  A  +  (\+r)=b       2.  A+b=l+r 

3.  !>x(l— r)=Z>       3.  D-±{\—r)=b       3.  Z>-v-fc=l—  r 
The  2d  and  3d  formulas  of  each  case  may  be  united  in  one; 

thus,  using   P   for   proceeds,  P=6x(l±r);  6=P-r-(l±r) ; 
r=P-i-6— 1.  or  1— P-t-6. 


360 


THE   PHILOSOPHY    OF    ARITHMETIC. 


Applications. — The  applications  of  Percentage  are  very  ex- 
tensive, owing  to  the  great  convenience  of  reckoning  by  the 
hundred  in  financial  transactions.  These  applications  are  of 
two  general  classes; those  not  including  the  element  of  time,  and 
those  which  include  this  element.  The  following  are  the  most 
important  of  these  two  classes  of  applications  : 


1st  Class. 

2d  Class. 

1. 

Profit  and  Loss. 

1. 

Simple  Interest. 

2. 

Stocks  and  Dividends. 

2. 

Partial  Payments. 

3. 

Premium  and  Discount. 

3. 

Discounting. 

4. 

Commission. 

4. 

Banking. 

5. 

Brokerage. 

5. 

Exchange. 

0. 

Insurance 

6. 

Equation  of  Payments. 

1. 

Taxes. 

7. 

Settlement  of  Accounts. 

3. 

Duties  and  Customs. 

8. 

Compound  Interest 

9. 

Stock  Investments. 

9. 

Annuities. 

The  different  cases  of  the  first  class  are  solved  as  in  pure 
percentage,  and  the  rules  are  almost  identical,  the  technical 
terms  being  substituted  for  base,  percentage,  etc.  The  solutions 
of  the  various  cases  of  the  second  class  are  somewhat  modified 
by  the  introduction  of  the  element  of  time.  The  development 
of  these  various  cases  would  occupy  too  much  space  for  this 
work,  and  moreover  does  not  constitute  a  part  of  the  philosophy 
of  arithmetic;  we  shall,  therefore,  give  only  a  single  chapter 
on  the  general  nature  of  Interest. 


CHAPTER  II. 

NATURE   OF  INTEREST. 

PERCENTAGE  embraces  two  general  classes  of  problems, 
those  that  involve  the  element  of  time,  and  those  that  do 

not  involve  this  element.  The  most  important  application  of 
percentage  into  which  this  element  enters  is  Interest ;  and  in- 
deed all  such  applications  may  be  embraced  under  this  general 

term. 

Interest  may  be  defined  as  money  paid,  or  charged  for  the 
use  of  money.  It  is  usually  reckoned  as  so  many  units  on  a 
hundred,  and  is  thus  included  under  the  general  process  of  Per- 
centage. The  sum  upon  which  interest  is  reckoned  is  called 
the  Principal,  in  distinction  from  the  interest  or  profit,  which 
is  subordinate  to  it.  The  sum  of  the  interest  and  principal  is 
called  the  Amount. 

Interest  is  either  Simple  or  Compound.  Simple  Interest  is 
that  which  is  reckoned  or  allowed  upon  the  principal  only, 
during  the  whole  time  of  the  loan.  Compound  Interest  is 
reckoned,  not  only  on  the  sum  loaned,  but  also  on  the  interest 
as  it  becomes  due.  Interest  unpaid  is  regarded  as  a  new  loan 
upon  which  interest  should  be  paid. 

Simple  Interest.— In  considering  the  subject  of  simple  inter- 
est, the  primary  object  is  to  find  the  interest  on  a  given  princi- 
pal for  a  given  time  and  rate.  Yarious  methods  have  been 
devised  for  the  solution  of  this  problem.  The  simplest  in 
principle  and  most  natural,  is  to  find  the  interest  for  one  year 
by  multiplying  the  principal  by  the  rate,  and  multiplying  this 
interest  by  the  time  expressed  in  years.  The  objection  to  this 
16  ( 361) 


362  THE   PHILOSOPHY   OF    ARITHMETIC. 

method  in  practice  arises  from  the  fact  that  the  time  is  often 
given  in  months  and  days,  which  frequently  reduce  to  an  incon- 
venient  fractional  part  of  a  year.  This  difficulty  has  led  to  a 
modification  of  the  rule  proposed  above,  which  is  known  as 
the  method  of  "aliquot  parts." 

The  importance  of  a  method  that  can  be  readily  applied  in  bus- 
iness; has  led  to  the  exercise  of  considerable  ingenuity  in  order 
to  discover  the  shortest  and  simplest  rule  in  practice.  The 
method  now  regarded  as  the  simplest  is  that  known  as  the 
"six  per  cent."  method.  It  is  based  on  the  rate  of  6%,  which  is 
the  usual  rate  in  this  country,  and  may  be  expressed  as  follows : 
Gall  half  the  number  of  months  cents,  and  one-sixth  of  the 
number  of  days  mills,  and  multiply  their  sum,  which  will  be 
the  interest  of  $1  for  the  rate  and  time,  by  the  principal. 
Another  way  of  stating  this  rule  is, — Regard  the  months  as 
cents,  and  one-third  of  the  days  as  mills,  and  multiply  their 
sum  by  one-half  of  the  principal.  For  short  periods  a  modi- 
fication of  the  rule,  which  may  be  popularly  expressed, — Mul- 
tiply dollars  by  days  and  divide  by  6000,  is  the  most  convenient 
in  practice,  and  is  very  generally  employed  by  business  men. 
There  are  also  many  other  methods  of  working  interest  which 
need  not  be  stated  here. 

The  general  method  of  finding  the  interest  of  a  principal  may 
be  expressed  in  a  general  formula  as  in  Percentage.  The  gen- 
eral formula  is  i=ptr,  which  is  readily  remembered  by  the  sen- 
tence which  it  suggests—  "  I  equals  Peter."  The  several  cases 
which  arise  in  interest  can  be  readily  derived  from  this  funda- 
mental formula.  These  several  rules  may  be  expressed  as  fol- 
lows: 

1.  i=ptr.  3.  t=i-i-pr. 

2.  p  =  i-i-tr.  4.  r  =  i-^pt. 

It  is  objected  to  the  "six  per  cent,  method,"  that  it  gives  too 
great  an  interest,  since  it  reckons  only  360  days  in  a  year ;  and 
it  has  been  suggested  that  to  compute  the  interest  on  a  loau  by 
this  method  would  be  to  take  usury,  and  in  some  states  would 


NATURE   OF   INTEREST.  363 

result  in  a  forfeiture  of  the  debt,  or  some  other  penalty.  This 
seems  like  putting  a  very  nice  point  on  the  matter,  though  it  is 
true  that  the  six  per  cent,  method  gives  a  little  more  interest 
than  when  we  reckon  365  days  to  the  year.  To  obtain  exact 
interest,  we  find  the  interest  for  the  years,  multiply  the  interest 
of  one  year  by  the  number  of  days,  and  divide  by  365,  and  take 
the  sum  of  the  two  results.  A  full  presentation  of  the  applica- 
tions of  interest  to  business  and  the  latest  methods  of  treatment 
may  be  found  in  the  author's  Higher  Arithmetic. 

Rates  of  Interest. — It  is  a  noteworthy  fact  that  the  propriety 
of  receiving  interest  for  the  use  of  money,  has  been  questioned. 
Indeed,  the  practice  has  been  censured  in  both  ancient  and 
modern  times  as  an  immorality  and  a  wrong  to  society.  It 
may  seem  that  so  absurd  a  notion  hardly  needs  a  passing  no- 
tice, for  it  is  clear  that  a  similar  objection  may  be  made  to  the 
charge  of  rents,  or  even  to  profits  of  any  kind.  A  capitalist 
may  invest  his  money  in  business  and  receive  a  certain  return 
for  it ;  and  if  he  chooses  to  let  some  one  else  invest  it  and  have 
the  care  of  such  investment,  it  is  clear  that  he  should  receive 
some  remuneration  for  surrendering  to  another  the  profit  he 
might  have  made  himself.  Again,  the  borrower  can  with  cap- 
ital secure  a  large  return  of  profit  in  business,  and  is  not  only 
entirely  willing  to  pay  for  the  use  of  such  capital,  but  is  in 
equity  under  obligations  to  do  so.  Interest  on  loans  is,  there- 
fore, a  benefit  to  both  the  borrower  and  lender;  and  should 
therefore  be  both  required  and  allowed. 

The  rate  of  interest  is  determined  strictly  by  the  principle 
of  competition.  When  the  capital  to  be  invested  exceeds  the 
demands  of  borrowers,  the  rate  of  interest  is  low ;  when  the 
demand  is  in  excess  of  the  capital,  the  rate  will  be  high.  The 
rate  will  vary  also  with  the  security  of  the  loan ;  thus  the  rate 
on  .landed  mortgages  is  usually  lower  than  on  property  less 
secure  and  certain,  and  consequently  state  loans  are  usually 
made  at  low  rates.  A  lender  assumes  that  he  must  be  paid 
something  for  the  risk  of  a  loan,  and  that  the  greater  the  risk 


364:  THE   PHILOSOPHY   OF    ARITHMETIC. 

the  greater  the  charge.  It  is  on  this  principle  that  high  inter 
est  is  often  said  to  be  synonymous  with  bad  security.  A  high 
rate  of  interest  may  also  be  due  to  large  profits  on  capital.  In 
a  community  where  the  returns  on  capital  are  large,  as  in  rich 
mining  districts  for  instance,  all  who  have  capital  would  desire 
to  invest,  and  consequently  the  difficulty  of  obtaining  a  loan 
would  increase  and  higher  rates  would  obtain.  In  such  cases 
the  opportunity  for  large  gains  by  the  capitalist  and  the  in- 
creased demand  by  the  borrower  would  both  conspire  to  increase 
the  rate  of  interest. 

The  rates  of  interest  have  usually  been  regulated  by  govern- 
ments. This  action  is  founded  upon  a  variety  of  reasons.  It 
has  been  argued  that  lenders  are  unproductive  consumers  of 
part  of  the  profit  which  is  produced  by  labor.  Such  a  notion 
leaves  out  of  sight,  however,  that  production  is  impossible 
without  capital,  and  that  capital  is  accumulated  and  employed 
with  a  view  to  profit.  It  is  also  held  that  if  the  state  does  not 
regulate  rates,  borrowers  will  be  open  to  fraud  and  extortion 
on  the  part  of  unprincipled  lenders.  This  is  the  principal  con- 
sideration in  favor  of  state  control  of  interest  rates ;  and  yet 
there  are  valid  if  not  unanswerable  objections  to  it.  It  is,  of 
course,  the  duty  of  the  government  to  protect  the  citizen  against 
usury  and  fraud ;  but  most  of  the  considerations  in  favor  of 
regulating  rates  of  interest  will  apply  to  the  regulation  of  the 
prices  of  food,  land,  wages,  etc.  It  seems  to  be  a  growing 
opinion  that  capital  should  seek  investment  at  rates  determined 
by  natural  laws  of  demand  and  supply,  as  the  prices  of  other 
property  are  regulated,  and  not  be  controlled  by  legislative  en- 
actment. 

Historical. — The  payment  of  interest  on  money  has  been 
the  custom  from  very  early  times.  We  learn  from  the  New 
Testament  that  it  was  paid  on  bankers'  deposits  in  Judea 
though  the  Jews  were  forbidden  by  the  laws  of  Moses  to  exact 
interest  from  one  another.  In  Europe,  interest  was  alternately 
prohibited  and  allowed,  the  church  being  generally  hostile  to 


NATURE   OF   INTEREST.  365 

the  practice.  In  Italy,  the  trade  in  money  was  recognized, 
and  the  custom  of  borrowing  and  lending  was  common.  In 
England,  it  was  first  sanctioned  by  the  Parliament  in  1546,  the 
rate  being  fixed  at  10  per  cent.;  but  in  1552  it  was  again  pro- 
hibited. Mary,  however,  borrowed  at  12  per  cent.,  which  ap- 
pears to  have  been  the  usual  rate  at  that  period  at  Antwerp. 
In  1571,  it  was  again  made  legal  at  10  per  cent.,  a  rate  at 
which  the  Scotch  Parliament  fixed  it  in  1587.  The  rate  fell  at 
the  beginning  of  the  seventeenth  century,  James  I.  having 
borrowed  in  Denmark  at  6  per  cent.  In  1G24,  it  was  reduced 
to  8  per  cent.;  in  1651,  to  6  per  cent.;  in  1724,  to  5  per  cent., 
at  which  legal  rate  it  remained  until  all  usury  laws  were  re- 
pealed, an  event  which  occurred  only  a  few  years  ago.  In 
1773,  it  was  limited  to  12  per  cent,  in  India.  In  1660,  the  rate 
in  Scotland  and  Ireland  was  from  10  to  12  per  cent.;  in  France 
7  per  cent.;  in  Italy  and  Holland  3  per  cent.;  in  Spain  from  10 
to  12  per  cent.;  in  Turkey  20  per  cent.;  but  the  East  India 
Company,  while  the  legal  rate  was  6  per  cent.,  continued  to 
borrow  at  4  per  cent. 

The  term  Usury,  meaning  the  "  use  of  a  thing,"  was  origi- 
nally applied  to  the  legitimate  profit  arising  from  the  use  of 
money,  and  meant  merely  the  taking  of  interest  for  money. 
Laws  were  established  in  various  countries  fixing  the  amount 
of  interest  or  usury,  and  the  evasion  of  these  laws  by  charging 
excessive  usury,  led  to  the  present  use  of  the  term.  By  the  old 
Roman  law  of  the  Twelve  Tables,  the  rate  of  interest  allowed 
as  legitimate  was  the  usura  centesima,  which  was  strictly  1 
per  cent,  a  month ;  and  has  been  supposed  by  some  to  have 
amounted  to  12,  and  by  others  to  10  per  cent,  a  year.  The 
Roman  laws  against  excessive  usury  were  frequently  renewed 
and  constantly  evaded,  and  the  same  is  true  of  other  countries. 
In  England,  during  the  reign  of  Henry  VIII.,  10  per  cent,  was 
allowed;  by  21  James  I.,  8  percent.;  by  12  Charles  II.f  8  per 
cent;  by  12  Anne,  5  per  cent.  Subsequently  to  the  passage 
of  the   latter  act,  the   usury  laws   were  relaxed  by  several 


366  THE    PHILOSOPHY    OF   ARITHMETIC. 

statutes,  and  they  were  ultimately  repealed  in  1854.  Any  rate 
of  interest,  however  high,  may  now  be  legally  stipulated  for, 
but  5  per  cent,  remains  the  legal  interest  recoverable  on  all 
contracts,  unless  otherwise  specified. 

Much  concern  has  been  shown  by  governments  in  attempt- 
ing to  fix  rates  of  interest,  and  prevent  usury.  The  legislation 
of  Solon  relieved  the  Athenian  mortgagors ;  and  during  many 
years  of  the  Roman  Republic,  the  regulation  of  loans,  the  limi- 
tation of  the  rate  of  interest,  and  the  relief  of  insolvent  debtors, 
formed  a  perpetual  topic  of  agitation,  and  finally  of  legislation. 
In  most  of  the  European  countries  the  administration  has 
busied  itself,  from  time  to  time,  in  fixing  rates  of  interest,  and 
in  denouncing  or  forbidding  usurious  bargains.  Such  legisla- 
tion has,  however,  proved  vain  ;  for  while  the  most  stringent 
laws  were  in  force,  high  rates  of  interest  on  loans  were  com- 
mon, the  law  being  incompetent  to  provide  against  evasion  of 
the  statute. 

The  legal  rate  of  the  United  States  government  is  6  per 
cent.  Each  State  fixes  its  own  rate,  and  attaches  its  special 
penalties  for  usury.  In  several  of  the  States  the  usury  laws 
have  been  repealed,  and  the  general  tendency  is  to  allow  an 
open  market  to  the  investment  of  capital. 

Origin  of  Methods. — The  importance  of  a  knowledge  of  the 
principles  of  interest,  discount,  etc.,  led  arithmeticians  to  notice 
these  subjects  at  an  early  day.  Interest  was  early  divided 
into  Simple  and  Compound.  Compound  Interest  was  properly 
called  usura,  and  was  rarely  practised  in  the  transactions  of 
merchants  with  each  other.  Stevinus  terms  compound  interest, 
interest  prouffitable,  or  celuy  qu'on  ajouste  au  capital,  whilst 
the  corresponding  discount  is  termed  interest  dommageable,  or 
celuy  qu'on  soubstrait  du  capital. 

Problems  in  simple  interest  were  by  Tartaglia  and  his  pre- 
decessors, solved  by  the  Rule  of  Three.  In  calculating  the 
interest  of  a  sum  from  one  day  to  another,  the  determination 
of  the  number  of  days  in  the  interval  seemed  somewhat  embar- 


NATURE   OF   INTEREST.  867 

rassing,  and  Tartaglia  giyes  a  rule  for  this  purpose  of  which  he 
seems  somewhat  proud.  In  passing  from  one  city  of  Italy  to 
another  an  additional  source  of  embarrassment  presented  itself 
in  the  different  days  on  which  the  year  was  supposed  to  com- 
mence, being  reckoned  at  Venice  from  the  1st  of  March,  at 
Florence  from  the  Annunciation  of  the  Virgin,  and  in  most 
other  cities  of  Italy  from  Christmas  day. 

Tartaglia  has  noticed  five  methods  of  finding  the  amount  of 
a  sum  of  money  at  compound  interest.  Suppose  the  question 
to  be  to  find  the  amount  of  L300  for  4  years  at  10  per  cent,  a 
capo  d'anno ;  the  first  method  is  by  the  following  four  state- 
ments : 

100  :  300  :  :  110  :  330 

100  :  330  :  :  110  :  363 

100  :  363  :  :  110  :  399^ 

100  :  399T3¥:  :  110  :  439T%. 
The  second  method  merely  replaces  100  and  110  by  10  and 
11  in  the  proportion ;  the  third,  which  is  his  own  method,  mul- 
tiplies 300  four  times  successively  by  11,  and  divides  the  last 
product  by  10,000  ;  the  fourth  consists  in  adding  four  suc- 
cessive tenths  to  the  principal;  the  last  in  calculating  the 
amount  for  L100,  and  then  finding  the  amount  of  L300,  or  any 
other  proposed  sum,  by  a  simple  proportion. 

With  the  exception  of  discount  at  compound  interest  and  its  ap- 
plication to  correct  in  part  the  conclusion  respecting  the  values 
of  annuities,  there  are  few,  if  any,  other  questions  of  compound 
interest  which  Tartaglia  and  his  contemporaries  can  be  said  to 
have  resolved.  A  very  natural  difficulty  arose  in  the  solution 
of  questions  of  this  kind  :  "  What  is  the  interest  of  100  for  6 
months,  interest  being  reckoned  at  the  rate  of  20  per  cent,  per 
annum  ?"  Lucas  di  Borgo  and  others  made  out  that  this  would 
be  10;  that  is,  they  calculated  that,  simple  interest  only  being 
allowed,  it  was  a  matter  of  indifference  into  how  many  por- 
tions of  time  the  whole  period  was  divided,  whether  into  months 
or  half-years. 

Lucas  di  Borgo  has  an  article  on  calculating  tables  of  inter- 


368  THE   PHILOSOPHY   OF   ARITHMETIC. 

est  in  which  he  speaks  of  their  great  utility,  thereby  showing 
that  such  tables  were  in  use  in  Italy,  although  no  work  of  that 
date  containing  them  is  known  to  be  extant.  The  first  com- 
pound interest  tables  now  known  are  those  which  are  presented 
by  Stevinus  in  his  arithmetic,  which  give  the  present  worth  of 
10,000,000  from  1  to  30  years,  in  sixteen  tables,  the  interest 
being  reckoned  successively  from  1  to  16  per  cent.,  and  in  eight 
other  tables,  where  the  interest  is  differently  reckoned,  accord- 
ing to  the  custom  of  Flanders. 

The  origin  of  the  various  modern  methods  of  calculating  in- 
terest is  not  known.  The  method  by  "aliquot  parts"  is  a  fav- 
orite rule  of  the  English  arithmeticians,  and  probably  originated 
with  them.  The  "  six  per  cent,  method"  has  been  attributed 
to  a  Mr.  Adams,  author  of  a  work  on  arithmetic.  The  partic- 
ular form  of  the  six  per  cent,  method  popularly  stated,  "multi- 
ply dollars  by  days  and  divide  by  6000,"  was  used  among 
business  men  before  it  was  introduced  into  any  arithmetic,  and 
is  presumed  to  have  had  its  origin  in  some  counting-house,  but 
it  is  not  known  where. 


SECTION   IV. 

THE  THEORY  OF  NUMBERS 


I.  Nature  of  the  Subject. 


II.  Even  and  Odd  Numbers. 


III.  Prime  and  Composite  Numbers. 


IY.  Perfect,  Imperfect,  etc.,  Numbers. 


V.  Divisibility  of  Numbers. 


VI.  Divisibility  by  the  Number  Seven. 


VII.  Properties  of  the  Number  Nine. 


CHAPTER  I. 

NATURE    OF    THE    SUBJECT. 

THE  Theory  of  Numbers,  as  generally  presented,  embraces 
the  classification  and  investigation  of  the  properties  of 
numbers.  This  subject  has  engaged  the  attention  and  enlisted 
the  talents  of  many  celebrated  mathematicians.  The  ancient 
writers,  who  did  little  for  the  development  of  arithmetic  as  a 
science  or  an  art,  spent  much  time  in  theorizing  upon  the  pro- 
perties of  numbers.  The  science  of  arithmetic  with  them  was 
mainly  speculative,  abounding  in  fanciful  analogies  and  mys- 
terious properties. 

Pythagoras  attributed  to  numbers  certain  mystical  properties, 
and  seems  to  have  conceived  the  idea  of  what  are  now  termed 
Magic  Squares.  Aristotle,  amongst  other  numerical  specula- 
tions, noticed  the  practice,  in  almost  all  nations,  of  dividing 
numbers  into  groups  of  tens,  and  attempted  to  give  a  philo- 
sophical explanation  of  the  cause.  The  earliest  regular  system 
of  numbers  is  that  given  by  Euclid  in  the  7th,  8th,  9th,  and 
10th  books  of  his  "  Elements,"  which,  notwithstanding  the 
embarrassing  notation  of  the  Greeks,  and  the  inadequacy  of 
geometry  to  the  investigation  of  numerical  properties,  is  still 
very  interesting,  and  displays,  like  all  other  parts  of  the  same 
celebrated  work,  that  depth  of  thought  and  accuracy  of  demon- 
stration for  which  its  author  is  so  eminently  distinguished. 

Archimedes,  also,  paid  particular  attention  to  the  powers  and 
properties  of  numbers.  His  tract,  entitled  "  Arenarius,"  con- 
tains a  method  of  multiplying  and  dividing  which  bears  a  con- 
siderable analogy  to  that  which  we  now  employ  in  multiplication 

(371) 


372  THE    PHILOSOPHY    OF    ARITHMETIC. 

and  division  of  powers,  and  which  some  modern  writers  have 
thought  inculcated  the  principles  of  our  present  system  of  loga- 
rithms. Before  the  invention  of  algebra,  however,  but  little 
progress  could  be  made  in  this  branch  of  the  science  ;  accord- 
ingly we  find  that  comparatively  few  principles  had  been  dis- 
covered until  the  time  of  Diophantus.  This  eminent  mathema- 
tician, who  is  the  author  of  the  most  ancient  existing  work  on 
the  subject  of  algebra,  presents  many  interesting  problems  in 
the  properties  of  numbers ;  but,  owing  to  the  difficulties  of  a 
complicated  notation  and  a  deficient  analysis,  little  progress 
was  made,  compared  with  the  advance  of  modern  times. 

From  the  time  of  Diophantus  the  subject  remained  unnoticed, 
or  at  least  unimproved,  until  Bachet,  a  French  analyst,  under- 
took the  translation  of  Diophantus  into  Latin.  This  work, 
which  was  published  in  1621,  contained  many  marginal  notes 
of  the  translator,  and  may  be  considered  as  presenting  the  first 
germs  of  our  present  theory.  These  were  afterward  consider- 
ably extended  by  Fermat,  in  his  posthumous  edition  of  the 
same  work,  published  in  1610,  which  contains  many  of  the  most 
elegant  theorems  in  this  branch  of  analysis ;  but  they  are  gen- 
erally left  without  demonstration,  which  he  explains  in  a  note 
by  saying  that  he  was  preparing  a  treatise  of  his  own  upon 
the  subject.  Legendre  accounts  for  the  omission  by  saying 
that  it  was  in  accordance  with  the  spirit  of  the  times  for  learned 
men  to  propose  problems  to  each  other  for  solution.  They 
generally  concealed  their  own  method  in  order  to  obtain  new 
triumphs  for  themselves  and  their  nation  ;  and  there  was  about 
this  time  an  especial  rivalry  between  the  English  and  French 
mathematicians.  Thus  it  has  happened  that  most  of  the  demon- 
strations of  Fermat  have  been  lost,  and  the  few  that  remain 
only  make  us  regret  the  more  those  that  are  wanting. 

The  most  of  these  theorems  remained  undemonstrated  until 
the  subject  was  again  renewed  by  Euler  and  Lagrange.  Euler, 
in  his  "Elements  of  Algebra,"  and  some  other  publications,  de- 
monstrated many  of  the  theorems  of  Fermat,  and  also  added 


NATURE   OF    THE   SUBJECT.  373 

some  interesting  ones  of  his  own.  Lagrange,  in  his  additions 
to  Euler's  Algebra  and  in  other  writings,  greatly  extended  the 
theory  of  numbers  by  the  discovery  of  many  new  properties. 
The  subject  has  received  its  largest  contributions,  however,  from 
the  hands  of  Gauss  and  Legendre. 

Legendre,  in  his  great  work,  "  Essai  sur  la  Theorie  des 
Nombres,"  was  the  first  to  reduce  this  branch  of  analysis  to  a 
regular  system.  Gauss,  in  his  "  Disquisitiones  Arithmetical, " 
opened  a  new  field  of  inquiry  by  the  application  of  the  proper- 
ties of  numbers  to  the  solution  of  binomial  equations  of  the 
form,  or1 — 1  —  0,  on  the  solution  of  which  depends  the  division  of 
the  circle  into  n  equal  parts.  This  solution  he  accomplished  in 
several  partial  cases;  whence  the  division  of  the  circle  into  a 
prime  number  of  equal  parts  is  performed  by  the  solution  of 
equations  of  inferior  degrees;  and  when  the  prime  number  is 
of  the  form  2n-fl  the  same  maybe  done  geometrically — a  prob- 
lem that  was  far  from  being  supposed  possible  before  the  publi- 
cation of  the  work  mentioned. 

The  most  celebrated  English  work  on  the  subject  is  that  of 
Peter  Barlow,  published  in  1811,  from  the  preface  of  which 
most  of  the  preceding  historical  facts  have  been  culled.  It  pre- 
sents a  clear  and  concise  statement  of  the  principles  of  the  sub- 
ject, and  contains  several  original  contributions,  among  which 
may  be  mentioned  a  demonstration  of  Fermat's  general  theorem 
on  the  impossibility  of  the  indeterminate  equation  xn±yn=zn, 
for  every  value  of  n  greater  than  2.  This  demonstration,  how- 
ever, has  been  tacitly  ignored  by  mathematicians;  and  the 
French  Institute  and  other  learned  societies  have  continued 
to  propose  the  problem  for  solution. 

Almost  every  modern  mathematician  of  eminence,  however, 
has  contributed  more  or  less  to  the  advancement  of  the  theory. 
In  the  collected  works  of  Euler,  Gauss,  Jacobi,  Cauchy, 
Dirichlet,  Lagrange,  Eiseastein,  Poinsot,  and  others,  numerous 
memoirs  on  the  subject  will  be  found ;  whilst  the  recent  mathe- 
matical journals  and  academical  transactions  contain  researches 


374  THE    PHILOSOPHY   OF    ARITHMETIC. 

in  the  same  field,  by  all  the  ablest  living  mathematicians.  One 
of  the  most  complete  treatises  on  the  subject  is  that  of  Frof. 
II.  J.  S.  Smith  in  the  article  entitled,  "Reports  on  the  Theory 
of  Numbers,"  which  commenced*  in  the ,  Transactions  of  the 
British  Association  for  1859.  It  embraces  a  lucid,  critical  his- 
tory of  the  subject,  rendered  doubly  valuable  by  copious  refer- 
ences to  the  original  sources  of  information. 

It  will  be  seen  from  this  brief  statement  that  the  subject  of 
the  theory  of  numbers  is  one  of  great  magnitude  and  difficulty 
requiring  the  application  of  the  principles  of  algebra  for  its  de- 
velopment. It  is,  therefore,  not  appropriate  to  treat  of  it  in 
this  work,  except  so  far  as  to  show  its  logical  relation  to  the 
general  divisions  of  the  science,  and  to  present  a  few  simple 
properties  that  may  be  readily  understood  by  means  of  the  or- 
dinary principles  of  arithmetic.  These  will  be  interesting  to 
young  arithmeticians,  and  perhaps  the  means  of  cultivating  a 
taste  for  a  more  thorough  study  of  the  subject. 

The  subjects  to  which  the  attention  of  the  reader  will  be 
briefly  directed  are  the  following: 

1.  Even  and  Odd  Numbers. 

2.  Prime  and  Composite  Numbers. 

3.  Perfect,  Imperfect,  etc.,  Numbers. 

4.  Divisibility  of  Numbers. 

5.  Divisibility  by  the  Number  Seven. 

6.  Properties  of  the  Number  Nine 


CHAPTER  II. 

EVEN   AND   ODD   NUMBERS. 

NUMBERS  have  been  divided  into  many  different  classes, 
founded  upon  peculiarities  discovered  by  investigating 
their  properties.  The  series  1,  2,  3,  4,  etc.,  is  called  the  series 
of  Natural  Numbers.  The  Natural  Numbers  are  classified 
with  respect  to  their  relation  to  the  number  two,  into  Odd  and 
Even  numbers.  They  are  also  divided  into  two  classes  with 
respect  to  their  composition,  called  Prime  and  Composite 
numbers.  Composite  Numbers  are  divided  into  two  classes, 
Perfect  and  Imperfect  numbers,  this  classification  being  based 
upon  the  relation  of  the  numbers  to  the  sum  of  their  factors. 
Imperfect  Numbers  are  also  divided  into  two  classes  with  re- 
spect to  the  numbers  being  greater  or  less  than  the  sum  of  their 
factors.  Numbers  which  are  equal  each  to  the  sum  of  the  di- 
visors of  the  other,  are  called  Amicable  Numbers.  A  few  re- 
marks will  be  made  on  each  one  of  these  classes. 

Of  the  various  classes  of  numbers,  the  simplest  and  most 
natural  division  is  that  of  Even  and  Odd  numbers.  This  di- 
vision is  founded  upon  the  relation  of  numbers  to  the  number 
2.  Even  numbers  are  those  which  are  multiples  of  2 ;  Odd 
numbers  are  those  which  are  not  multiples  of  2.  In  the  series 
of  natural  numbers  the  increase  is  by  a  unit;  in  the  series  of 
even  numbers  the  scale  of  increase  is  dual.  The  former  arise 
from  counting  by  l's,  beginning  with  the  unit;  the  latter  in 
counting  by  2's,  beginning  with  the  duad.  The  even  numbers 
are  divided  into  the  oddly  even  numbers,  2,  6,  10,  14,  etc.;  and 
the  evenly  even  numbers,  4,  8,  12,  l<i,  etc.  The  odd  numbers 
are  divided  into  the  evenly  odd  numbers  1,  5,  9,  13,  etc;  and 
the  oddly  odd  numbers,  3,  7,  11,  15,  etc. 

(375) 


376  THE   PHILOSOPHY    OF   ARITHMETIC. 

The  formula  for  the  even  numbers  is  2n ;  the  formula  for  the 
odd  numbers  is  2ra-fl.  In  the  oddly  even  numbers  n  is  an  odd 
number  ;  in  the  evenly  even  numbers  n  is  an  even  number.  In 
the  evenly  odd  numbers  n  is  even ;  in  the  oddly  odd  numbers 
n  is  odd.  The  evenly  odd  numbers  are  of  the  form  4n-f  1 ;  the 
oddly  odd  numbers  are  of  the  form  in-\-  3. 

There  are  many  interesting  principles  relating  to  even  and 
odd  numbers,  a  few  of  which  will  be  stated. 

1.  Every  prime  number  except  2  is  an  odd  number. 

2.  The  differences  of  the  successive  square  numbers  produce 
the  odd  numbers. 

3.  The  sum  or  difference  of  two  even  numbers  or  two  odd 
numbers  is  an  even  number. 

4.  The  sum  or  difference  of  an  even  number  and  an  odd  num- 
ber is  odd. 

5.  The  sum  of  any  number  of  even  numbers  is  even  ;  the 
sum  of  an  even  number  of  odd  numbers  is  even,  and  the  sum 
of  an  odd  number  of  odd  numbers  is  odd. 

6.  The  product  of  two  even  numbers  is  even;  of  two  odd  num- 
bers is  odd  ;  of  an  even  number  and   an   odd  number  is  even. 

7.  The  quotient  of  an  even  by  an  odd  number,  when  exact, 
is  even;  the  quotient  of  an  odd  by  an  odd,  when  exact,  is  odd; 
the  quotient  of  an  even  by  an  even,  when  exact,  is  either  even 
or  odd. 

8.  An  odd  number  is  not  exactly  divisible  by  an  even  num- 
ber, and  the  remainder  is  odd. 

9.  If  an  even  number  is  not  exactly  divisible  by  an  even 
number,  its  remainder  is  even. 

10.  If  an  even  number  is  not  exactly  divisible  by  an  odd 
number,  then  when  the  quotient  is  even  the  remainder  is  even, 
and  when  the  quotient  is  odd,  the  remainder  is  odd. 

11.  If  an  odd  number  is  not  exactly  divisible  by  an  odd 
number,  then  when  the  quotient  is  odd  the  remainder  is  even, 
and  when  the  quotient  is  even  the  remainder  is  odd. 

12.  If  an  odd  number  divides  an  even  number,  it  will  also 


EVEN    AND   ODD   NUMBERS.  377 

divide  one-half  of  it ;  if  an  even  number  be  divisible  by  an  odd 
number,  it  will  be  divisible  by  double  that  number. 

13.  Any  power  of  an  even  number  is  even  ;  and  conversely 
the  root  of  an  even  number  which  is  a  complete  power  is  even 

14.  Any  power  of  an  odd  number  is  odd  ;  and  conversely 
the  root  of  an  odd  number  which  is  a  complete  power  is  odd. 

15.  The  sum  or  difference  of  any  complete  power  and  its  root 
is  even. 

These  principles  can  be  readily  proved  by  the  ordinary  meth- 
ods of  arithmetical  reasoning.  To  illustrate,  take  the  third 
principle,  the  reasoning  of  which  is  as  follows :  Two  even 
numbers  are  each  a  number  of  2's,  hence  their  sum  will  be  the 
sum  of  two  different  numbers  of  2's,  which  must  be  a  number 
of  2's,  and  their  difference  will  be  the  difference  between  two 
different  numbers  of  2's,  which  is  also  a  number  of  2's.  In  add- 
ing two  odd  numbers  we  will  have  a  number  of  2's-f  1,  added 
to  another  number  of  2's-fl,  which  will  give  us  a  number  of 
2's +2,  or  an  exact  number  of  2's,  etc. 

The  simplest  method  is  by  using  the  general  notation  of  al- 
gebra. Thus  in  the  given  principle,  these  two  even  numbers 
will  be  represented  by  2n  and  2nf ;  their  sum  will  be  2n+2nf, 
or  2  (n-\-nr),  which  is  of  the  form  of  2n,  and  is  thus  even ;  their 
difference  will  be  2n — 2nr,  or  2(n — n'),  which  is  of  the  form  of 
2n,  and  is  even.  The  two  odd  numbers  are  of  the  form  2n  -f-1 
and  2nf+  1,  and  their  sum  is  2  (n-ftt'+l),  which  is  of  the  form 
of  2n,  and  even ;  their  difference  is  2n — 2nf,  or  2  (n — nr)}  which  is 
evidently  even.  All  the  other  principles  may  be  demonstrated 
in  a  similar  manner. 


CHAPTER  III. 

PRIME  AND  COMPOSITE  NUMBERS. 

TIIE  most  celebrated  classification  of  numbers  is  that  of  Prime 
and  Composite.  This  classification  is  with  respect  of  their 
formation  by  multiplication  or  the  possibility  of  their  being  re- 
solved into  factors.  The  Composite  number  is  one  which  can 
be  produced  by  the  multiplication  of  other  numbers  ;  the  Prime 
number  is  one  which  cannot  be  produced  by  the  multiplication 
of  other  numbers.  The  distinction  may  be  regarded  as  having 
reference  to  the  dependence  or  independence  of  their  existence. 
The  composite  number  is  regarded  as  deriving  its  existence 
from  other  numbers  which  make  it ;  the  prime  number  does 
uot  derive  its  being  from  any  other  numbers,  but  is  indepen- 
dent and  self-existent. 

Perhaps  no  subject  in  arithmetic  has  received  more  attention 
from  mathematicians  than  that  of  Prime  and  Composite  Numbers. 
The  object  has  been  to  discover  some  general  method  of  find- 
ing prime  numbers,  and  of  determining  whether  a  given  num- 
oer  is  prime  or  composite.  Such  a  method,  though  laboriously 
sought  for  by  the  best  mathematical  minds,  has  not,  beyond  a 
certain  limit,  been  discovered. 

The  problem  of  ascertaining  prime  numbers  was  discussed 
as  far  back  as  the  days  of  Eratosthenes,  a  mathematician  of 
Alexandria,  distinguished  also  as  having  first  conceived  the 
plan  of  measuring  the  earth.  He  invented  a  method  of  obtain- 
ing primes  by  excluding  from  the  series  of  natural  numbers 
those  that  are  not  prime,  and  thus  discovering  those  that  are. 
This  method  consisted  in  inscribing  the  series  of  odd  numbers 
upon  parchment,  and  then  cutting  out  the  composite  numbers, 

(378) 


PRIME    AND   COMPOSITE   NUMBERS.  379 

and  leaving  the  primes.  The  parchment,  with  its  holes,  resem- 
bled a  sieve ;  hence  the  method  is  called  Eratosthenes1  sieve. 
Ilis  method  may  be  illustrated  as  follows: 

Suppose  we  write  the  series  of  odd  numbers  from  1  to  99  in- 
clusive. Since  the  series  increases  by  2,  the  third  term  from 
3  is  3  +  3x2,  which  is  divisible  by  3;  hence  every  third  term 
is  divisible  by  3,  and  is  therefore  composite.  In  a  similar 
manner  we  see  that  every  fifth  term  after  5  is  divisible  by  5, 
and  therefore  composite  ;  and  every  seventh  term  after  1  is  di- 
visible by  7,  and  therefore  composite.  Cutting  out  these  com- 
posite numbers,  we  have  all  the  prime  numbers  below  100.  By 
this  method,  assisted  by  some  mechanical  contrivance,  Vega 
computed  and  published  a  table  of  prime  numbers  from  1  to 
400,000. 

This  method  is,  however,  very  tedious  and  inconvenient,  and 
mathematicians  have  earnestly  sought  for  properties  of  prime  and 
composite  numbers  to  guide  them  in  ascertaining  primes.  The 
following  principles  are  useful  in  discovering  or  determining 
prime  numbers: 

1.  All  prime  numbers  except  2  are  odd,  and  consequently 
terminate  with  an  odd  digit.  The  converse  of  this,  that  all  odd 
numbers  are  prime,  is  not,  however,  true. 

2.  All  prime  numbers,  except  2  and  5,  must  terminate  with 
1,  3,  7,  or  9  ;  all  other  numbers  are  composite.  This  is  the 
series  of  odd  digits  with  the  omission  of  5,  since  any  number 
terminating  with  5,  can  be  divided  by  5  without  a  remainder. 

3.  Every  prime  number,  except  2,  if  increased  or  diminished 
by  1,  is  divisible  by  4.  In  other  words,  every  prime  number, 
except  2,  is  of  the  form  4n  db  1.  This  will  admit  of  demonstra- 
tion. 

4.  Every  prime  number,  except  2  and  3,  if  increased  or  di- 
minished by  1,  is  divisible  by  6.  In  other  words,  every  prime 
number,  except  2  and  3,  is  of  the  form  6w  ±  1.  This  may  also 
be  demonstrated. 

5.  Every  prime  number,  except  2,  3,  and  5,  is  a  measure  of 


380  THE   PHILOSOPHY    OF   ARITHMETIC. 

the  number  expressed,  in  common  notation,  by  as  many  l's  as 
there  are  units,  less  one,  in  the  prime  number.  Thus,  7  is  a 
measure  of  111,111 ;  and  13  of  111,111,111,111. 

6.  Every  prime  number,  except  2  and  5,  is  contained  with- 
out a  remainder  in  the  number  expressed  in  the  common  nota- 
tion by  as  many  9's  as  there  are  units,  less  one,  in  the  prime 
number  itself.  Thus,  3  is  a  measure  of  99  ;  7  of  999,999;  and 
13  of  999,999,999,999. 

7.  Three  prime  numbers  cannot  be  in  arithmetical  progression, 
unless  their  common  difference  is  divisible  by  6  ;  except  3  be 
the  first  prime  number,  in  which  case  there  may  be  three  prime 
numbers  in  such  progression,  but  in  no  case  can  there  be  more 
than  three. 

8.  This  last  principle  is  generally  true,  and  may  be  stated 
as  follows:  There  cannot  be  n  prime  numbers  in  arithmetical 
progression  unless  their  common  difference  be  divisible  by 
2.3.5.7.  11...  n;  except  the  case  in  which  n  is  the 
first  term  of  the  progression,  in  which  case  there  may  be  n  such 
numbers,  but  not  more. 

Though  we  have  no  general  method  for  finding  prime  num- 
bers, there  are  several  ways  of  detecting  whether  an  assigned 
number  is  or  is  not  a  prime.  Several  remarkable  formulas  have 
been  discovered  which  contain  a  large  number  of  prime  num- 
bers. The  formula  x2-\-x+4l,  by  making  successively  x=0, 
1,  2,  3,  4,  etc.,  will  give  a  series  41,  43,  47,  53,  61,  71,  etc.,  the 
first  forty  terms  of  which  are  prime  numbers.  This  formula  is 
mentioned  by  Euler  in  the  Memoirs  of  Berlin,  1772.  Of  the 
two  formulas  x2+ #+17,  and  2j?2+29,  the  former  gives  seven- 
teen of  its  first  terms  primes,  and  the  latter  twenty-nine.  For- 
mat asserted  that  the  formula  2"»4-l  is  always  a  prime  when 
m  is  taken  any  term  in  the  series  1,  2,  4,  8,  16,  etc.;  but  Euler 
found  that  2^+1=641  x  6,700,417  is  not  a  prime. 

One  of  the  most  celebrated  theorems  for  investigating  primes 
is  that  discovered  by  Format  and  known  as  Ftsr mat's  Theorem. 
The  theorem  may  be  stated  thus:  If  p  be  a  prime, the  (p — l)th 


PRIME  AND  COMPOSITE    NUMBERS.  381 

power  of  every  number  prime  to  p  will,  when  diminished 
by  unity,  be  exactly  divisible  by  p.  Expressed  in  algebraic 
language,  we  have  the  theorem  P>— *— -1,  is  a  multiple  of  p  when 
p  and  P  are  prime  to  each  other.  Thus,  256— 1  is  exactly 
divisible  by  7. 

Fermat  is  said  to  have  been  in  possession  of  a  proof  of  the 
theorem,  though  Euler  was  the  first  to  publish  its  demonstra- 
tion. Euler's  first  demonstration  was  a  very  simple  one,  and 
is  that  usually  given  in  the  text-books.  Amongst  the  other 
demonstrations  of  the  theorem,  those  given  by  Lagrange  are 
highly  esteemed. 

It  has  been  demonstrated  by  Legendre  (Essai  sur  la  Theorie 
des  Nombres),  that  every  arithmetical  progression,  of  which  the 
first  term  and  common  difference  are  prime  to  each  other,  con- 
tains an  infinite  number  of  prime  numbers.  It  has  been  also 
shown  by  him  that  if  n  represents  any  number,  then  will  the 
formula 

N 


h.logx— 1.08366 
represent  the  number  of  prime  numbers  that  are  less  than  n, 
very  nearly. 

Another  celebrated  theorem  is  that  invented  by  Sir  John 
Wilson,  known  as  Wilson's  Theorem.  This  theorem  may  be 
stated  as  follows  :  The  continued  product,  increased  by  unity, 
of  all  the  integers  less  than  a  given  prime,  is  exactly  divisible 
by  that  prime.  The  algebraic  formula  which  expresses  the  the- 
orem, 1  +  1.2.3...  (n— 1),  is  divisible  by  n  ,  n  being  a  prime 
number.     Thus  1  +  1.2.3.4.5.  6=721,  is  exactly  divisible 

by  7. 

This  theorem  was  first  demonstrated  by  Lagrange  ;  his  pro 
cess  of  reasoning,  as  might  be  expected,  was  very  ingenious. 
It  was  afterward  demonstrated  by  Euler,  and  finally  by  Gausb, 
who  extended  the  theorem  by  proving  that  uThe  product  of 
all  those  numbers  less  than,  and  prime  to,  a  given  number, 
a±l,  is  divisible  by  a ;"  the  ambiguous  sign  being  — ,  when  a 


382  THE    PHILOSOPHY    OF    ARITHMETIC. 

is  of  the  form  pm,  or  2pm,  p  being  any  prime  number  greater 
than  2  ;  and,  also,  when  a=4;  but  positive  in  all  other  cases. 

Wilson's  Theorem  furnishes  us  with  an  infallible  rule,  in 
theory,  for  ascertaining  whether  a  given  number  be  a  prime  or 
not ;  for  it  evidently  belongs  exclusively  to  those  numbers,  as  it 
fails  in  all  other  cases ;  but  it  is  of  no  use  in  a  practical  point 
of  view,  on  account  of  the  great  magnitude  of  the  product  even 
for  a  few  terms. 

In  the  later  works  on  the  Theory  of  Numbers  it  is  demon- 
strated that,  No  algebraical  formula  can  represent  prime 
numbers  only.  It  is  also  shown  that,  The  number  of  prime 
numbers  is  infinite.  The  latter  proposition  is  evident  a 
priori;  the  former  was  pretty  nearly  evident  from  induction 
before  it  received  a  rigid  demonstration. 

The  distribution  of  prime  numbers  does  not  follow  any  known 
law;  but  for  a  given  interval  it  is  found  that  the  number  of 
primes  is  generally  less  the  higher  the  beginning  of  the  interval 
is  taken.  The  whole  number  of  primes  below  10,000  is  1,230; 
between  10,000  and  20,000  it  is  1,033;  between  20,000  and 
30,000  it  is  983  ;  between  90,000  and  100,000  it  is  879.  The 
largest  prime  which  had  been  verified  when  Barlow  wrote,  is 
231— 1  =  2,147,483,647,  which  was  found  by  Euler. 

The  term  prime  is  also  applied  to  a  species  of  numbers  called 
complex  numbers,  first  suggested  by  Gauss  in  1825.  Accord- 
ing to  this  theory,  a  complex  integer  is  of  the  form  a  -j_  by/Z^{ 
in  which  a  and  b  denote  ordinary  (real)  integers.  The  product 
a2-f&2,  of  a  complex  number  a+ &x/— ^  ana"  its  conjugate, 
a — &v/^T,  is  called  its  norm,  and  is  denoted  by  the  symbols 
N(a  +  &x/^T),  N(a — b^/^i).  The  four  associative  numbers, 
a  +  &v/^T,  ay/~—i—b1  — a—by/^\,  and  — a^/^i  +  b,  as  well 
as  their  respective  conjugates,  have  all  the  same  norm.  A  com- 
plex number  is  said  to  be  prime  when  it  admits  of  no  divisor 
except  itself,  its  associatives,  and  the  four  units,  1,  — 1,  v/— i, 
and  — \/— !■  Many  of  the  higher  theorems,  such  as  that  of 
Format,  may  be  extended  to  the  system  of  complex  numbers. 


CHAPTER   IV. 

PERFECT,   IMPERFECT,   ETC.,  NUMBERS. 

HAYING  separated  numbers  into  their  factors,  the  human 
mind,  ever  active  in  the  attempt  to  discover  the  new,  be- 
gan to  compare  the  sum  of  the  factors  or  divisors  of  numbers 
with  the  numbers  themselves,  and  thus  discovered  certain  re- 
lations which  gave  rise  to  three  new  classes  of  numbers.  In 
some  cases  it  was  seen  that  a  number  was  just  equal  to  the 
sum  of  all  of  its  divisors,  not  including  itself,  and  such  num- 
bers were  called  Perfect  Numbers.  Numbers  not  possessing 
this  property  were  called  Imperfect  Numbers ;  and  were  divided 
into  two  classes,  Defective  and  Abundant,  according  as  they 
were  greater  or  less  than  the  sum  of  their  divisors. 

Pushing  the  comparison  still  further,  it  was  also  discovered 
that  some  numbers  were  reciprocally  equal  to  their  divisors; 
and  this  relation  was  so  intimate  that  such  numbers  were  re- 
garded as  friendly  or  Amicable  Numbers.  These  several  classes 
will  be  formally  defined  in  this  chapter.  Perfect  and  Imperfect 
numbers  were  known  by  the  ancient  Greek  mathematicians,  but 
their  properties  have  been  developed  by  the  mathematicians  of 
modern  times.  Amicable  Numbers  were  first  investigated  by 
the  Dutch  mathematician  Van  Schooten,  who  lived  from  1581  to 
1646. 

A  Perfect  Number  is  one  which  is  equal  to  the  sum  of  all  its 
divisors,  except  itself;  thus,  6=1+2+3;  28=1+2+4+7  +  14 
An  Imperfect  Number  is  one  which  is  not  equal  to  the  sum 
of  all  its  divisors.  Imperfect  Numbers  are  Abundant  or  De- 
fective. An  Abundant  Number  is  one  the  sum  of  whose  di- 
visors exceeds  the  number  itself;    as,  l+2+3+6+9>18.     A 

(383) 


384  THE    PHILOSOPHY    OF    ARITHMETIC. 

Defective  Number  is  one  the  sum  of  whose  divisors  is  less 
than  the  number  itself;  as,  1+2+4+8  <  16. 

Every  number  of  the  form  (2'1—1)  (2n—  1),  the  latter  factor 
being  a  prime  number,  is  a  perfect  number.  The  only  values 
of  n  yet  found,  which  make  2H — 1  a  prime  are  2,  3,  5,  7,  13,  17, 
19,  and  31 ;  there  are,  therefore,  only  ten  perfect  numbers 
known.  Substituting  2  for  n  in  the  formula,  we  have  2(22 — 1) 
=6,  the  first  perfect  number;  the  second  is  22(23 — 1)=28. 
The  first  eight  perfect  numbers  are,  6,  28.  4%.  8128.  33550336, 
8589869056,  137438691328,  2305843008139952128.  Each 
number,  as  is  seen,  ends  in  6  or  28. 

The  difficulty  in  finding  perfect  numbers  consists  in  finding 
primes  of  the  form  of  2n — 1.  The  greatest  prime  number,  ac- 
cording to  Barlow,  yet  ascertained,  is  2S1 — 1  =  2147483647,  dis- 
covered by  Euler  ;  and  the  last  of  the  above  perfect  numbers, 
which  depends  upon  this,  is  the  greatest  perfect  number  known 
at  present,  and  Barlow  remarks  that  it  is  probably  the  greatest 
that  will  ever  be  discovered ;  for,  as  they  are  merely  curious 
without  being  useful,  it  is  not  likely  that  any  person  will 
attempt  to  find  one  beyond  it.  An  author  of  an  arithmetic  gives 
two  other  numbers  which  are  said  to  be  perfect,  2417851639228- 
158837784576,  9903520314282971830448816128,  but  I  do  not 
know  his  authority. 

Two  numbers  are  called  Amicable  when  each  is  equal  to  the 
sum  of  the  divisors  of  the  other  ;  thus,  284  and  220.  The  for- 
mulas for  finding  amicable  numbers  are  A=2n+ld  and  B= 
2n+ibc,  in  which  n  is  an  integer,  and  b,  c,  and  d  are  prime 
numbers  satisfying  the  following  conditions:  1st,  6=3  x  2* — 1 ; 
2d,  c=6x2n— 1;  3d,  d=18x22n—  1.  If  we  make  n=l,  we 
find  6=5,  c=ll,  and  d=71;  substituting  these  in  the  above 
formulas,  we  have  ^4=4x71=284,  and  £=4x5x11^220,  the 
first  pair  of  amicable  numbers.  The  next  two  pairs  are 
17296,  18416,  and  936358,  9437056. 

The  first  pair,  220  and  284,  were  found  by  E.  Yan  Schooten, 
with  whom  the  name  amicable  appears  to  have  originated,  though 


PERFECT,    IMPERFECT,   ETC.,  NUMBERS.  385 

Rudolphus  and  Descartes  were  previously  acquainted  with 
this  property  of  certain  numbers.  A  formula  for  amicable 
numbers  was,  in  fact,  given  by  Descartes,  and  afterwards  gen- 
eralized by  Euler  and  others. 

Figurate  Numbers. — Figurate  Numbers  are  numbers  formed 
from  an  arithmetical  progression  whose  first  term  is  unity,  and 
common  difference  integral,  by  taking  successively  the  sum  of 
the  first  two,  the  first  three,  the  first  four,  etc.,  terms  of  the 
series ;  and  then  operating  on  the  new  series  in  the  same  man- 
ner as  in  the  original  progression  in  order  to  obtain  a  second 
series,  and  so  on. 

For  example,  take  the  series  of  natural  numbers  in  which 
the  common  difference  is  1,  as  repre- 
ented  by  A  in  the  margin:  then  the     A,  1-2  -  3  -  4  -  5  -  6  -  7 

T^lTAIOI^      01       Oft 

series  B,  derived  as  stated  above,  will  „'  ~\Z  n'on'oe  ~  56  ~  g4 
be  figurate  numbers ;  series  C,  derived  p'  1-5-15-35-10-126-210 
as  above  from  series  B  and  series  D, 

derived  from  series  C,  will  be  figurate  numbers.  Other  series 
could  be  obtained  by  beginning  with  any  other  arithmetical 
series  whose  first  term  is  1,  and  common  difference  an  integer. 
Thus,  the  series  derived  from  the  progression  1,  3,  5,  7,  9,  etc., 
is  1,  4,  9,  16,  25,  etc. 

A  more  general  method  of  conceiving  figurate  numbers  is  to 
regard  them  as  a  series  of  numbers,  the  general  term  of  eacb 
series  being  expressed  by  the  formula, 

n(n+\)(n  +  2)(n+S)     ....     (n-f-m) 
1.2.3.4         .         .         .      (m-fiy 
in  which  m  represents  the  order  of  the  series,  and  n  represents 
the  place  of  the  required  term. 

Series  of  figurate  numbers  are  divided  into  orders;  when  m 
=  0,  the  series  is  of  the  1st  order;  when  m=  1,  the  series  is 
of  the  2d  order;  when  m  =  2,  it  is  of  the  3d  order,  etc. 

By  regarding  m  equal  to  0  in  this  formula,  and  substituting 
successive  numbers  1,  2,  3,  etc.,  for  n,  it  will  be  seen  that  the 
general  term  is  n,  and  we  find  that  the  figurate  series  of  the 
first  order  is  the  series  of  natural  numbers,  1,  2,  3,  4,  etc.,  n. 
25 


386  THE    PHILOSOPHY   OF    ARITHMETIC. 

By  regarding  m  equal  to   1,  the  general  term  of  the  series 

becomes  — — -— »  and  substituting  the  successive  values  of  n, 
1  .  2 

1,  2,  3,  etc.,  we  find  the  terms  to  be  1,  3,  6,  10,  15,  21,  28,  etc., 

which  is  the  series  of  figurate  numbers  of  the  second  order. 

Ir  a  similar  manner  we  find  the  general  term  of  the  figurate 

scries  of  the  3d  and  4th  orders  to  be  respectively, 

n(n  +  1)0  +  2)  n(n  +  l)Q+2)  (n  +  3). 

and  — 


1.2.3  1.2.3.4 

from  which  we  can  readily  derive  those  series.  These  several 
series  of  figurate  numbers  are  the  same  as  those  represented  in 
the  margin  above. 

One  of  the  most  remarkable  properties  of  the  series  of  figu- 
rate numbers  is  that,  if  the  nth  term  of  a  series  of  any  order  be 
added  to  the  (n-\-l)ih  term  of  the  series  of  the  preceding 
order,  the  sum  will  be  equal  to  the  (ra-f  l)th  term  of  the  series 
of  the  given  order.  Thus,  in  the  series  marked  C,  if  we  add 
the  second  term,  4,  to  the  third  term,  6,  in  series  B,  we  shall 
have  the  third  term,  10,  of  series  C ;  the  third  term  of  series  C 
plus  the  fourth  term  of  series  B  equals  the  fourth  term  of  series 
C,  etc. 

If  we  begin  with  a  series  of  l's,  all  of  the  series  of  figurate 
numbers  may  be  deduced  in  succession  by  the  application  of 
th;s  principle. 

Orders  op  Figurate  Numbers. 

Series  of  l's  1,  1,  1, 
1st  order  1,  2,  3, 
2d  order  1,  3,  6, 
3d  order  1,  4,  10, 
4th  order  1,  5,  15,  35,  70,  126,  210,  330,  495,  715 
5th  order  1,  6,  21,  56,  126,  252,  462,  792,  1287,  2002 
6th  order  1,  7,  28,  84,  210,  462,  924,  1716,  3003,  5005 
1th  order  1,  8,  36,  120,  330,  792,  1716,  3432,  6435,  11440 
By  inspecting  these  series,  it  will  be  seen  that  tho  values 


1, 

1, 

1, 

1, 

1, 

1, 

1 

4, 

5, 

6, 

7, 

8, 

9, 

10 

10, 

15, 

21, 

28, 

36, 

45, 

55 

20, 

35, 

56, 

84, 

120, 

165, 

220 

PERFECT,    IMPERFECT,   ETC.,  NUMBERS.  387 

ead  diagonally  upward  are  the  numerical  coefficients  of  the 
terms  in  the  development  of  (a-\-b)  with  an  exponent  corre- 
sponding to  the  order  of  the  series.  It  is  said  that  it  was  this 
principle  which  gave  rise  to  a  complete  investigation  of  the 
subject  of  iigurate  numbers. 

In  speaking  of  defining  figurate  numbers  by  giving  the  form 
of  each  of  the  orders,  Barlow  remarks  that  it  is  more  simple  to 
deduce  the  generation  of  figurate  numbers  from  their  form  than 
to  deduce  their  form  from  their  generation.  The  principle 
given  above,  showing  the  relation  of  the  terms  of  two  succes- 
sive orders  of  figurate  numbers,  is  ascribed  to  Format,  and  is 
considered  by  him  as  one  of  his  most  interesting  propositions. 

Polygonal  Numbers  are  figurate  numbers  which  represent 
the  sides  of  polygons.  The  second  series  of  figurate  numbers, 
i,  3,  6,  10,  etc.,  are  called  triangular 

numbers,  because  the  number  of  units  •  •      • 

that  they  express  can  be  arranged  in       •     •     • 
the  form  of  a  triangle.     If  we  take 
the  series  1,  3,  5,  7,  9,  etc.,  in  which  ... 

the  common  difference  is  2,  we  obtain  .... 

the  figurate  series,  1,  4,  9,  16,  25,  etc.,  which  are  called  square 
numbers,  because  they  can  be  arranged  in  a  square.  The 
series  1,  4,  T,  10,  etc.,  in  which  the  common  difference  is  3, 
gives  the  series  1,  5,  12,  22,  etc.,  which  are  called  pentagonal 
numbers,  because  they  can  be  arranged  in  the  form  of  a  penta- 
gon. In  a  similar  manner  we  obtain  hexagonal,  heptagonal, 
jctagonal,  etc.,  numbers.  It  will  be  noticed  that  the  number 
of  the  sides  of  the  polygon  which  they  represent  is  always  two 
greater  than  the  common  difference  of  the  series  from  which 
they  were  derived.  Common  differenced 
When  the  common  Triangular  numbers 
,._,  „     ,,        Common  diflerence=2: 

difference     of     the     Square  numbers 
series  in  arithmeti-    Common  diflerenee=3 

i  •        •       Pentagonal  numbers 

cal   progression  is  ° 

1,  the  sums  of  the  terms  give  the  triangular  numbers;  when 


1, 

2 

3, 

4, 

5, 

6 

1, 

3,' 

6, 

10, 

15, 

21 

1, 

3, 

5, 

7, 

9, 

11 

1, 

4, 

9, 

16, 

2o, 

36 

1, 

4, 

7, 

10, 

13, 

16 

1. 

o, 

12, 

35, 

51 

888  THE   PHILOSOPHY   OP   ARITHMETIC. 

the  common  difference  is  2,  the  sums  of  the  terms  are  the  square 
numbers;  when  the  difference  is  3,  the  sums  are  the  pentagonal 
numbers,  and  so  on. 

These  numbers  are  called  polygonal  from  possessing  the  pro- 
perty that  the  same  number  of  points  may  be  arranged  in  the 
form  of  that  polygonal  figure  to  which  it  belongs.  Thus  the 
pentagonal  numbers  5,  12,  22,  35,  51,  etc.,  may  be  severally 
arranged  in  the  form  of  a  pentagon.  Thus,  5  points  will  form 
one  pentagon ;  12  points  will  form  a  second  pentagon  enclos- 
ing the  former ;  22,  a  third  pentagon  enclosing  both  of  the 
former,  etc. 

The  following  property  of  polygonal  numbers  was  discovered 
by  Ferinat:  Every  number  is  either  a  triangular  number  or 
the  sum  of  two  or  three  triangular  numbers;  every  number  is 
either  a  square  number,  or  the  sum  of  two,  three,  or  four 
square  numbers;  every  number  is  either  a  pentagonal  number 
or  the  sum  of  two,  three,  four,  or  five  pentagonal  numbers  ; 
etc.  This  property  is  generally  true,  although  it  has  been 
demonstrated  for  only  triangular  and  square  numbers.  All 
the  other  cases  still  remain  without  demonstration,  notwith- 
standing the  researches  of  many  of  the  ablest  mathematicians. 
Format  himself,  however,  as  appears  from  one  of  his  notes  on 
Diophantus,  was  in  possession  of  the  demonstration,  although 
it  was  never  published,  which  circumstance  renders  the  theorem 
still  more  interesting  to  mathematicians,  and  the  demonstration 
of  it  more  desirable. 

Pyramidal  Numbers  are  those  which  represent  the  number 
of  bodies  that  can  be  arranged  in  pyramids.  They  are  formed 
by  the  successive  sums  of  polygonal  numbers  in  the  same  man- 
ner as  the  polygonal  numbers  are  formed  from  arithmetical 
progressions.  The  Triangular  Pyramidal  numbers  are  the 
series  of  figurate  numbers  derived  from  the  series  of  triangular 
numbers.  Thus,  from  the  triangular  numbers  1,  3,  6,  10,  15, 
etc.,  we  have  the  triangular  pyramidal  numbers  1,  4,  10,  20, 
etc.  The  Square  Pyramidal  numbers  are  derived  from  the 
square  numbers. 


CHAPTER  V. 

DIVISIBILITY   OF    NUMBERS. 

TN  factoring  a  composite  number,  we  divide  successively  by 
exact  divisors  of  the  number  till  we  obtain  a  quotient  which 
is  a  prime  number.  In  order  to  know  by  what  numbers  to 
divide,  it  is  convenient  to  have  some  tests  of  divisibility,  other- 
wise it  would  be  necessary  to  try  several  numbers  until  we  hit 
upon  one  which  is  exactly  contained.  There  are  certain  laws 
which  indicate,  without  the  test  of  actual  division,  whether  a 
number  is  divisible  by  a  given  factor,  some  of  which  are  simple 
and  may  be  readily  applied  The  investigation  of  these  laws 
of  the  relations  of  the  factors  of  numbers  to  the  numbers  them- 
selves, gives  rise  to  a  subject  known  as  the  Divisibility  of  Num- 
bers. 

The  laws  for  the  divisibility  of  numbers,  as  usually  presented, 
embrace  the  conditions  of  divisibility  by  the  numbers  2,  3,  4, 
etc.,  up  to  12.     These  laws  may  be  stated  as  follows: 

1.  A  number  is  divisible  by  2  when  the  right-hand  term  is 
zero  or  an  even  digit.  For,  the  number  is  evidently  an  even 
number,  and  all  even  numbers  are  divisible  by  2. 

2.  A  number  is  divisible  by  3  when  the  sum  of  the  numbers 
denoted  by  its  digits  is  divisible  by  3.  It  will  be  shown  here- 
after that  every  number  is  a  multiple  of  9,  plus  the  sum  of  its 
digits;  hence,  since  3  is  a  factor  of  9,  the  number  is  divisible 
by  3  when  the  sum  of  the  digits  is  divisible  by  3. 

3.  A  number  is  divisible  by  4,  when  the  two  right-hand  terms 
are  ciphers,  or  when  they  express  a  number  which  is  divisible 
by  4.     If  the  two  right-hand  terms  are  ciphers,  the  number 

(389) 


390  THE    PHILOSOPHY    OF    ARITHMETIC. 

equals  a  number  of  hundreds,  and  since  100  is  divisible  by  4, 
any  number  of  hundreds  is  divisible  by  4.  If  the  number  ex- 
pressed by  the  two  right-hand  digits  is  divisible  by  4,  the  num- 
ber will  consist  of  a  number  of  hundreds,  plus  the  number  ex- 
pressed by  the  two  right-hand  digits ;  and  since  both  of  these 
are  divisible  by  4,  their  sum,  which  is  the  number  itself,  is 
divisible  by  4. 

4.  A  number  is  divisible  by  5,  when  its  right-hand  term  is 
0  or  5.  If  the  right-hand  term  is  0,  the  number  is  a  number 
of  times  10 ;  and  since  10  is  divisible  by  5,  the  number  itself 
is  divisible  by  5.  If  the  right-hand  term  is  5,  the  entire  num- 
ber will  consist  of  a  number  of  tens,  plus  5 ;  and  since  both 
of  these  are  divisible  by  5,  their  sum,  which  is  the  number 
itself,  is  divisible  by  5. 

5.  A  number  is  divisible  by  6,  when  it  is  even  and  the  sum 
of  the  digits  is  divisible  by  3.  Since  the  number  is  even,  it  is 
divisible  by  2,  and  since  the  sum  of  the  digits  is  divisible  by  3, 
the  number  is  divisible  by  3,  and  since  it  contains  both  2  and  3 
it  will  contain  their  product,  3x2,  or  6. 

6.  A  number  is  divisible  by  7,  when  the  sum  of  the  odd  nu- 
merical periods,  minus  the  sum  of  the  even  numerical  periodsy 
is  divisible  by  7.  The  law  for  the  divisibility  by  7  is  perhaps 
of  not  so  much  practical  importance  as  the  others,  being  not 
quite  so  readily  applied,  but  it  is  of  too  much  scientific  interest 
to  be  omitted  from  the  series.  Its  demonstration  will  be  given 
in  the  following  chapter. 

7.  A  number  is  divisible  by  8,  when  the  three  right-hand  terms 
are  ciphers,  or  when  the  number  expressed  by  them  is  divisible 
by  8,  If  the  three  right-hand  terms  are  ciphers,  the  number 
equals  a  number  of  thousands;  and  since  1000  is  divisible  by  8, 
any  number  of  thousands  is  divisible  by  8.  If  the  number  ex- 
pressed by  the  three  right-hand  digits  is  divisible  by  8,  the 
entire  number  will  consist  of  a  number  of  thousands,  plus  the 
number  expressed  by  the  three  right-hand  digits  (thus  17368 
=  17,000  +  368);  and  since  both  of  these  parts  are  divisible  by 
8,  their  sum,  which  is  the  number  itself,  is  divisible  by  8. 


DIVISIBILITY    OP    NUMBERS.  891 

8.  A  number  is  divisible  by  9,  when  the  sum  of  the  digits  is 
divisible  by  9.  This  law  is  derived  from  showing  that  a  num- 
ber may  be  resolved  into  two  parts,  one  part  being  a  multiple 
of  9  and  the  other  the  sum  of  the  digits.  A  complete  demon- 
stration is  presented  on  a  subsequent  page,  to  which  the  reader 
is  referred. 

9.  A  number  is  divisible  by  10,  when  the  unit  term  is  0.  For, 
such  a  number  equals  a  number  of  lens,  and  any  number  of  tens 
is  divisible  by  10;  hence  the  number  is  divisible  by  10. 

10.  A  number  is  divisible  by  11,  when  the  difference  between 
the  sums  of  the  digits  in  the  odd  places  and  in  the  even  places 
is  divisible  by  11,  or  when  the  difference  is  0.  This  law 
is  derived  by  showing  that  a  number  may  be  resolved  into  two 
parts,  one  part  being  a  multiple  of  11,  and  the  other  part  con- 
sisting of  the  sum  of  the  digits  in  the  odd  places,  minus  the 
sum  of  the  digits  in  the  even  places.  A  complete  demonstra- 
tion will  be  presented  on  a  subsequent  page. 

11.-4  number  is  divisible  by  12,  when  the  sum  of  the  digits 
is  divisible  by  3  and  the  number  expressed  by  the  two  right- 
hand  digits  is  divisible  by  4.  For,  since  the  sum  of  the  digits 
is  divisible  by  3,  the  number  is  divisible  by  3,  and  since  the 
number  expressed  by  the  two  right-hand  digits  is  divisible  by 
4,  the  number  is  divisible  by  4 ;  hence,  since  the  number  is 
divisible  by  both  3  and  4,  it  is  divisible  by  their  product,  or  12. 

These  laws  arc  simple,  and,  with  the  exception  of  those  re- 
lating to  the  numbers  7,  9,  and  11,  readily  applied.  The  laws  of 
dividing  by  9  and  11  present  some  interesting  points,  which  will 
be  formally  discussed.  It  will  be  noticed,  upon  examining  text- 
books on  arithmetic,  and  also  works  on  the  theory  of  numbers, 
that  the  law  of  divisibility  by  7  is  omitted.  Apparently  efforts 
^ere  made  to  discover  such  a  law,  for  several  writers  give 
some  special  rules  for  dividing  by  7  ;  but  it  would  seem  that 
no  general  law  was  known  to  them.  In  the  principle  as  above 
presented,  this  hiatus  is  filled  up  by  a  law  not  quite  so  simple 
as  that  for  the  other  numbers,  but  still  of  scientific  interest,  if 


392  THE    PHILOSOPHY    OF    ARITHMETIC. 

Qot  of  much  practical  value.  Besides  the  law  given,  there  are 
several  other  laws,  interesting  as  showing  the  development  of 
the  subject,  and  which  we  therefore  present.  The  methods  of 
demonstration  are  similar  to  those  used  in  proving  the  divisi- 
bility of  numbers  by  9  and  11;  indeed,  one  of  the  laws  from 
which  the  others  were  derived  was  discovered  by  the  applica- 
tion of  that  method  to  the  number  7.  I  shall  therefore  first 
present  the  demonstration  of  divisibility  by  9  and  11,  and  then 
state  and  demonstrate  the  laws  relating  to  the  number  T. 

Divisibility  by  Nine. — The  law  of  divisibility  by  nine  has 
been  known  for  a  long  time.  By  whom  it  was  discovered  has 
not  been  ascertained.  Its  application  to  testing  the  correctness 
of  the  work  in  the  fundamental  rules,  called  proof  by  "  casting 
out  nines,"  has  been  attributed  to  the  Arabs.  The  law,  as  pre- 
viously stated,  is  that  a  number  is  divisible  by  nine  when  the 
sum  of  the  digits  is  divisible  by  nine.  This  principle  depends 
on  a  more  general  law  which  will  be  first  stated,  and  then  the 
law  of  exact  division,  as  well  as  some  other  interesting  princi- 
ples, will  be  drawn  from  it. 

1.  A  number  divided  by  9  leaves  the  same  remainder  as  the 
sum  of  the  digits  divided  by  9. 

This  theorem  can  be  demonstrated  both  arithmetically  and 
algebraically.  We  will  first  present  the  arithmetical  demonstra- 
tion. If  we  take  any  number,  as  6854,  and  analyze  it,  as  in 
the  margin,  r       ^_  4 

we  will  see  fio*4  I  50=5x10  =5x  (9+l)=5x9  +5 
that  it  con-  )    800=8x100  =8x    (99  +  l)=8x  99  +8 

sists  of  two  L  6000=6  x  1000=6  x  (999+l)=6  X  999+6 

parts*    the  Multiple  of  9 Sum  of  digits 

first  part  a    •"'  6854  =  5x9+8x99+6x999  +  4+5+8+6 
multiple  of  9,  and  the  second  part  the  sum  of  the  digits. 

The  first  part  is  evidently  divisible  by  9,  hence  the  only  re- 
mainder that  can  arise  from  dividing  a  number  by  9  will  be 
equal  to  the  remainder  arising  from  dividing  the  sum  of  the 
digits  by  9.     When  the  sum  of  the  digits  is  exactly  divisible 


DIVISIBILITY    OF    NUMBERS.  393 

by  9,  it  is  evident  that  the  number  itself  is  exactly  divisible 
by  9,  which  proves  the  theorem.  From  this  theorem  the  fol- 
lowing principles  may  be  readily  inferred: 

2.  A  number  is  exactly  divisible  by  9  when  the  sum  of  its 
digits  is  divisible  by  9. 

3.  The  difference  between  any  number  and  the  sum  of  its 
digits  is  divisible  by  9. 

4.  A  number  divided  by  9  gives  the  same  remainder  as  any 
one  formed  by  changing  the  order  of  the  figures. 

5.  The  difference  between  two  numbers,  the  sums  of  whose 
digits  are  equal,  is  exactly  divisible  by  9. 

The  fundamental  theorem  may  also  be  demonstrated  algebra- 
ically as  follows:  Let  a,  b,  c,  d,  etc.,  represent  the  digits  of 
any  number,  and  r  the  radix  of  the  scale,  that  is,  the  number 
of  units  in  a  group ;  then  every  number  may  be  represented 
by  formula  (1)  below.  If  we  now  subtract  b,  c,  d,  etc.,  from 
one  part  of  this  expression,  and  add  them  to  another  part,  it 
will  not  change  the  value,  and  we  shall  have  formula  (2) ;  and 
factoring,  we  obtain  formula  (3). 

(1).  #=a+&r+cr2+dr3+erM-etc. 

(2).  JV=&r  —  b+cr2— c+dr3— d+er4— e,  etc.+a  +  b  +  c+d+e 
+etc. 

(3).  N=b (r-1)  +c (r2-l)  +d (r3-l)  +e (r*-l)  +etc.  +a 
-\-b\-c+d+e+etc. 

Now,  r— 1,  r2— 1,  r3— 1,  etc.,  etc.,  are  all  divisible  by  r— 1 ; 
hence  the  only  remainder  which  can  arise  from  dividing  the 
number  by  r— 1,  will  occur  from  dividing  a+b-\-c-\-d+ etc.,  by 
r—  1;  that  is,  any  number  divided  by  r—  1  leaves  the  same 
remainder  as  the  sum  of  the  digits  divided  by  r— 1.  In  our 
decimal  scale  r=10,  hence  r— 1=9;  and  hence  any  number 
divided  by  9  leaves  the  same  remainder  as  the  sum  of  the  digits 
divided  by  9.  This  law  is  the  basis  of  some  very  interesting 
properties,  and  also  of  the  proof  of  the  fundamental  rules  called 
"casting  out  nines." 

Divisibility  by  Eleven.— The  law  of  the  divisibility  of  num- 
17* 


394  THE    PHILOSOPHY    OF    ARITHMETIC. 

bers  by  11  is  quite  similar  to  that  of  9.  This  might  have  been 
anticipated,  as  they  each  differ  from  the  basis  of  the  scale  by 
unity,  the  former  being  a  unit  below  and  the  latter  a  unit  above 
the  base.  The  law,  as  previously  stated,  is  that  a  number  i& 
divisible  by  11  when  the  difference  between  the  sum  of  the 
digits  in  the  odd  places  and  the  even  places  is  divisible  by  11. 
This  principle  depends  upon  a  more  general  one,  which  will  first 
be  stated,  and  then  this,  as  well  as  some. other  interesting  prin- 
ciples, will  be  derived  from  it. 

1.  Every  number  is  a  multiple  of  11,  plus  the  sum  of  the 
digits  in  the  odd  places,  minus  the  sum  of  the  digits  in  the 
even  places.  This  principle  may  be  demonstrated  both  arith- 
metically and  algebraically.  We  will  first  give  the  arithmetical 
proof.     If  we  take  any  number,  as  G5478,  and  analyze  it  as  in- 

Q  I    Q 

70=      7x10=    7x(H— 1)=    7x11—7 

65478=-       400=    4x100=    4 x  (994-1)=    4x99+4 

5000=  5x  1000=5  x(  1001— 1)=5X  1001—5 

.C0OOO=GxlO00O=Gx(9999+l)=Cx  9999+6 

Sum  of  Sum  of 
Multiples  of  11. odd  digits.       even  digits., 

/.  65478=7x11+4x99+5x1001+6x9999  +  "8+4+6"  —  5+T~ 
dicated,  we  shall  see  that  it  consists  of  two  parts;  the  first 
being  a  multiple  of  11,  and  the  second  consisting  of  the  sum 
of  the  digits  in  the  odd  places,  minus  the  sum  of  the  digits  in 
the  even  places.  The  first  part  is  evidently  divisible  by  11 ; 
hence  the  only  remainder  that  can  arise  from  dividing  a 
number  by  11  will  be  equal  to  the  remainder  arising  from 
dividing  the  difference  between  the  sums  of  the  digits  in  the 
odd  places  and  the  even  places  by  11.  When  this  difference  is 
exactly  divisible  by  11,  it  follows  that  the  number  itself  is 
divisible  by  11.  When  the  sum  of  the  digits  in  the  even  places 
is  greater  than  the  sum  in  the  odd  places,  we  take  the  difference, 
divide  by  11,  and  subtract  the  remainder  from  11  to  find  the 
true  remainder.  The  reason  for  this  will  appear  from  the 
above  demonstration.  From  this  theorem  the  following  prin- 
ciples can  be  readily  inferred : 

2.  A  number  is  exactly  divisible  by  11,  when  the  sum  of  the 


DIVISIBILITY    OF    NUMBERS.  395 

digits  in  the  odd  places  is  equal  to  the  sum  of  the  digits  in  the 
even  places. 

3  A  number  is  exactly  divisible  by  11,  when  the  difference 
between  the  sums  of  the  digits  in  the  odd  places  and  the  even 
places  is  a  multiple  of  11. 

4.  A  number  increased  by  the  sum  of  the  digits  in  the  even 
places  and  diminished  by  the  sum  of  the  digits  in  the  odd 
places,  is  exactly  divisible  by  11. 

5.  The  excess  of  IVs  in  any  number  is  not  changed  by  add- 
ing any  multiple  of  11  to  the  sum  of  the  digits  of  either  order. 

The  algebraic  demonstration  of  this  property  is  as  follows: 
Taking  the  same  formula  as  for  the  number  9,  we  add  b  and 
then  subtract  6,  we  subtract  c  and  'hen  add  c,  etc.,  the  formula 
becoming  (2)  below,  being  the  same  in  value  as  the  first, 
but  changed  in  form.     Then,  factoring,  we  have  (3). 

(1).  N=a-~br+cr'  Ldr'+er4+etc. 

(2).  2?=br+b+cr2— c+dr3+d+er*— e+etc.+a— 6+c— d+et 
etc. 

(3).^=6(r+l)+c(r2-l)  +  d(r3+l)+e(r4-l)+etc.+(a4- 
c+e-f  etc.)  —  (6+oH-etc.) 

Now  r+1,  r2—  1,  r3+l,  etc.,  are  each  divisible  by  r+1; 
hence  the  only  remainder  that  can  arise  from  dividing  this 
number  by  r+1  must  arise  from  dividing  (a+c+e+etc.) — 
(6+d+etc.)  by  r+1 ;  that  is,  by  dividing  the  difference  of  the 
sum  of  the  digits  in  the  even  places  subtracted  from  the  sum 
of  the  digits  in  the  odd  places  by  r+ 1 .  In  the  decimal  scale, 
r=10,  and  r+l=ll;  hence  we  see  that  any  number  divided 
by  11  leaves  the  same  remainder  as  the  difference  of  the  sum 
of  the  digits  in  the  even  places,  subtracted  from  the  sum  of  the 
digits  in  the  odd  places  does  when  divided  by  11.  When  this 
difference  is  exactly  divisible  by  11,  the  number  itself  is  divisi- 
ble, which  proves  the  principle  of  the  divisibility  by  11.  This 
principle  may  also  be  used  for  the  proof  of  the  fundamental 
rules,  but  not  quite  so  conveniently  as  that  of  the  number  9. 


CHAPTER  VI. 

THE   DIVISIBILITY   BY   SEVEN. 

THE  Divisibility  of  Numbers,  as  presented  by  different 
authors,  embraces  the  conditions  of  divisibility  by  the 
numbers  2,  3,  etc.,  up  to  12,  with  the  omission  of  the  num- 
ber 7.  This  omission  leads  us  to  inquire  whether  there  is 
any  general  law  for  the  divisibility  of  numbers  by  7.  A  few 
of  our  text-books  present  some  special  truths  in  regard  to  this 
subject,  among  which  are  the  following : 

1.  A  number  is  divisible  by  7  when  the  unit  term  is  one-half 
or  one-ninth  of  the  part  on  the  left.  Thus  21,  42,  63,  126,  and 
91,  182,  273,  etc. 

2.  A  number  is  divisible  by  7  when  the  number  expressed 
by  the  two  right-hand  terms  is  five  times  the  part  on  the  left, 
or  one-third  of  it.  Thus  525,  840,  1995,  and  602,  903,  3612, 
etc. 

3.  A  number  consisting  of  not  more  than  two  numerical 
periods  is  divisible  by  7  when  these  periods  are  alike.  Thus 
45045,  235235,  506506,  etc.,  are  divisible  by  7. 

There  are,  however,  some  general  laws  for  the  divisibility 
by  7,  which  seem  to  have  been  overlooked  by  most  writers  on 
the  theory  of  numbers,  and  which,  though  of  not  much  practical 
importance,  are  interesting  in  a  scientific  point  of  view.  The 
first  and  least  simple  of  these  laws  is  as  follows : 

1.  A  number  is  divisible  by  7,  when  the  sum  of  once  the 
first,  or  units  digit,  3  times  the  second,  2  times  the  third,  6 
times  the  fourth,  4  limes  the  fifth,  5  times  the  sixth,  once  the 
seventh,  3  times  the  eighth,  etc.,  is  divisible  by  7.     It  will  be 

(396) 


THE   DIVISIBILITY    BY    SEVEN.  397 

seen  that  the  series  of  multipliers  is  1,  3,  2,  6,  4,  5.  To  illus- 
trate the  law,  take  the  number  7935942,  and  we  have  for  the 
sum  of  the  multiples  of  the  digits,  1  x  2+3  x  4+2  x  9+6x5+4  x 
3+5  x  9  +  1  x  7=12G,  which  is  exactly  divisible  by  7  ;  and  if  we 
divide  the  number  itself  by  7,  we  find  there  is  no  remainder. 
Assuming  this  principle — it  will  be  demonstrated  on  page 
398 — we  can  derive  several  other  principles  of  divisibility 
from  it. 

In  this  law  we  see  that  the  second  half  of  the  series  of  mul- 
tipliers, 6,  4,  5,  equals  respectively  7  minus  the  first  half,  1,  3,  2; 
hence,  instead  of  adding  the  multiples  of  the  second  series,  G,  4, 
5,  we  may  subtract  the  respective  multiples  of  the  terms  of  the 
second  period  by  the  first  series  of  multipliers,  1,  3,  2,  which 
will  give  rise  to  the  following  principle  : 

2.  A  7iumber  is  divisible  by  7,  when  the  number  arising 
from    the    sum  of  once  the   first  digit,   3   times  the  second, 

2  times  the  third,  minus  the  sum  of  the  same  multiples  of  the 
next  three  digits,  plus  the  sum  of  the  same  multiples  of  the 
next  three  digits,  etc.,  is  divisible  by  7. 

It  will  be  seen  that  the  series  of  multipliers  is  1,  3,  2,  the 
first  products  additive,  the  second  products  subtractive,  etc. ; 
the  odd  numerical  periods  being  additive  and  the  even  periods 
subtractive.     If  we  take  the  number  5439728,  we  have  1x8  + 

3  x  2+2  x  7—1 X  9—3  x  3—2  x  4+1  x  5=7,  which  is  divisible  by 
7.  Upon  trial  we  find  the  original  number  is  also  exactly  di- 
visible by  7. 

This  second  principle  may  also  be  stated  thus:  A  number  is 
divisible  by  7  when  the  sum  of  the  multiples  expressed  by  the 
numbers,  1,  3,  2,  of  the  terms  of  the  odd  numerical  periods, 
minus  the  sum  of  the  same  multiples  of  the  terms  of  the  even 
numerical  periods,  is  divisible  by  7. 

Now,  if  we  add  exact  multiples  of  7  to  the  multiples  of  the 
terms  which  are  united  in  the  test  of  divisibility,  it  will  not 
change  the  remainder.  Thus,  taking  the  number  5439728,  if 
we  add  7  X  2  to  3  x  2,  we  have  10  x  2,  or  20  ;  and  adding  98  x  7 


398  THE    PHILOSOPHY    OF    ARITHMETIC. 

,0  2x7  we  have  100x7,  or  700;  hence  we  may  use  in  place 
>f  1x8+3x2+2x7,  8+20+700,  or  728,  the  first  numerical 
period ;  and  in  the  same  way  it  may  be  shown  that  we  may 
use  the  second  period  subtractively  in  the  test,  etc.  Hence 
from  Principle  2  we  may  derive  the  following  principle : 

3.  A  number  is  divisible  by  7,  when  the  sum  of  the  odd  nu- 
merical periods,  minus  the  sum  of  the  even  numerical  periods, 
is  divisible  by  7. 

To  illustrate,  take  the  number  5,643,378,762;  we  have  for 
the  sum  of  the  odd  numerical  periods  762+643=1405 ;  for  the 
sum  of  the  even  periods,  378+5=383;  the  difference  is  1022, 
which  is  exactly  divisible  by  7  ;  and  if  we  divide  the  number 
itself  by  7,  we  find  that  there  is  also  no  remainder. 

If  we  apply  the  same  reasoning  to  Principle  1,  by  which  we 
derived  Principle  3  from  Principle  2,  we  shall  derive  from  it  the 
following  principle : 

4.  A  number  is  divisible  by  7,  when  the  sum  of  the  numbers 
denoted  by  the  double  numerical  periods  is  divisible  by  7. 
Thus,  in  the  number  5,643,378,762,  we  have  5,643+378,762= 
384,405,  which  is  divisible  by  7,  and  the  number  is  also  divisi- 
ble by  7. 

The   first  principle,  from  which  I  have   derived  the  other 
three,  may  be  demonstrated  arithmetically  and  algebraically. 
Let  us  take  any  number  as  98765432  and  analyze  it  thus : 

1X2 
3x(7+3)=  3x7+3x3 

4x(98+2)=  4x98+2x4 

5X  (994+6)=        5x994+6x5 
6x  (9996+4)=      6x9996+4x6 
7X  (99995+5)=    7x991)95+5x7 
8000000=  8X1000000=  8x  (999999+1)=  8x999999+1x8 
90000000=9  X  10000000=9  X  (9999997+ 3) =9x9999997+3x9 
Here  98765432=a  multiple  of  7  plus  once  the  1st  term,  plus 
three  times  the  second  term,  plus  two  times  the  third  term,  plus 
six  times  the  fourth  term,  plus  four  times  the  fifth  term,  plus 
five  times  the  sixth  term,  plus  once  the  seventh  term,  plus  three 
times  the  eighth  term.     Hence  the  only  remainder  that  can  occur 
must  arise  from  dividing  the  sum  of  the  multiples  of  the  terms 


2= 

30= 

3X10= 

400= 

4x100= 

5000= 

5X1000= 

60000= 

6x10000= 

700000= 

7x100000= 

THE    DIVISIBILITY    BY    SEVEN.  399 

by  7 ;  hence  when  the  sum  of  these  multiples  is  divisible  by  7, 
the  number  is  divisible  by  7,  which  proves  the  principle. 

The  second  principle,  which  is  readily  derived  from  the  first, 
may  be  demonstrated  independently,  as  follows: 


2= 

1X2 

30= 

3X10= 

3X(7+S)= 

3x7+3x3 

400= 

4x100= 

4x  (984-2)= 

4x934-2x4 

5000= 

5X1000= 

5x(1001— 1)= 

5x1001—1x5 

60000= 

6X10000= 

6  x(  10003—3)= 

6x10003—3x6 

700000= 

7x100000= 

7x  (100002—2)  = 

7x100002—2x7 

8000000=  8x1000000=  8 X  (9999994-1)=  8x9999994-1x8 
90000000=9  x  10000000=9  X  (99999974-3) =9x99999974-3x0 

Here  98765432=a  multiple  of  7,  plus  once  the  first  digit, 
plus  three  times  the  second,  plus  twice  the  third,  minus  once  the 
fourth,  minus  three  times  the  fifth,  minus  twice  the  sixth,  plus 
once  the  seventh,  plus  three  times  the  eighth.  Hence  the  only 
remainder  that  can  occur  must  arise  from  dividing  the  difference 
between  the  additive  and  subtractive  multiples  of  the  digits  by 
7  ;  therefore,  when  this  difference  is  divisible  by  7,  the  number 
is  divisible  by  7,  which  proves  the  principle.  When  the  sum 
of  the  subtractive  multiples  of  the  digits  is  greater  than  the 
sum  of  the  additive,  we  take  the  difference,  divide  by  7,  and 
subtract  the  remainder  from  7  to  find  the  true  remainder. 

To  demonstrate  the  third  principle,  take  any  number,  as  7,946,- 
321,675  and  analyze  it,  and  it  will  be  seen  to  consist  of  parts 
which  are  multiples  of  7,  plus  the  periods  in  the  odd  places, 
minus  the  periods  in  the  even  places. 

675=  675 

321000=        321 X  (1001— 1)=  321x1001—321 

940000000=    946x  (9999994-1) =946 X 999x10014-946 

7000000000=7  X  (1000000001— l)=7x  999001 X 1001—    7 

Multiples  of  7. Odd  periods.  Even  periods. 

321x10014-946x9999994-7x1000000001  4-"675+946  -  3214-7 
Now  1001  is  a  multiple  of  7,  999999  is  999  times  1001,  and 
1000000001  is  also  a  multiple  of  1001,  and  if  we  continue  the 
number  to  still  higher  periods,  we  shall  find  a  constant  series 
of  multiples  of  1001,  alternately  1  more  and  1  less  than  the 
number  represented  by  one  unit  of  the  period.  Ilence 
7,946,321,675  is  composed  of  the  sum  of  three  multiples  of  7, 
plus  (6754-946)  —  (321 -f  7),  or  the  difference  betweeD  the  sums 


7946321675= 


400  THE   PHILOSOPHY    OP   ARITHMETIC. 

of  the  even  and  odd  periods.  The  first  part  is  evidently  divisi- 
ble by  7,  therefore  the  divisibility  of  the  number  depends  on 
the  divisibility  of  the  difference  of  the  sums  of  the  odd  and 
even  periods;  and  when  this  difference  is  divisible  by  7,  the 
number  itself  must  be  divisible  by  7,  which  proves  the  prin- 
ciple. 

From  this  demonstration,  we  can  immediately  derive  the  fol- 
lowing principle,  more  general  than  the  one  stated  and  from 
which  that  may  be  derived: 

5.  Any  number  divided  by  7  gives  the  same  remainder  as  is 
obtained  when  the  sum  of  the  odd  numerical  periods,  minus  the 
sum  of  the  even  numerical  periods,  is  divided  by  7.  If  the  sum 
of  the  even  periods  is  the  greater,  we  find  the  difference,  divide 
by  7,  and  subtract  the  remainder  from  7  for  the  true  remainder. 

This  investigation  leads  to  a  still  more  general  principle  of 
divisibility,  derived  from  the  fact  that  1001,  which  may  be  con- 
sidered as  the  basis  of  the  above  demonstration,  is  the  product 
of  7,  11,  and  13;  hence  what  we  have  just  proved  for  7,  is  also 
true  of  1 1  and  13.  The  most  general  form  of  the  principle  then 
is  as  follows: 

6.  Any  number  divided  by  7,  11,  or  13  gives  the  same  re- 
mainder  as  is  obtained  when  the  sum  of  the  odd  numerical 
periods,  minus  the  sum  of  the  even  numerical  periods,  is  divided 
by  7,  11,  or  13  respectively. 

A  special  truth  growing  out  of  this  general  principle,  had 
been  previously  given  in  the  rule  that  any  number  of  not  more 
than  two  periods,  when  those  two  periods  are  alike,  is  divisible 
by  7, 1 1,  or  13.  All  such  numbers,  on  examination,  will  be  found 
to  be  multiples  of  1001,  and,  of  course,  divisible  by  its  factors. 
It  may  seem  surprising  that  those  who  were  familiar  with 
this  special  truth,  and  were  thus  on  the  very  brink  of  a  dis- 
covery, did  not  extend  it  and  reach  the  general  law  above  pre- 
sented. 

The  fourth  Principle,  which  was  derived  from  the  first,  maj 
also  be  demonstrated  independently  by  a  method  similar  to 
that  used  in  proving  the  third  Principle.     The  algebraic  demon- 


THE   DIVISIBILITY   BY   SEVEN.  401 

stration  of  Principle  1,  which  is  the  foundation  of  the  other 
principles,  is  as  follows:  Take  the  same  general  formula  as  used 
in  demonstrating  the  divisibility  by  9  and  11,  add  and  subtract 
36,  32c,  33d,  etc.,  and  the  formula  is  readily  reduced  to  the  form 
of  (5). 

(1).     if=a+6r+cr2+dr3+er4+/r5+yr6+/ir7+etc. 

(2).     JV=6r-36+cr2-32c+ir3— 33d+er4— 3Vf/r5— 35/,  etc 
+a+36+9c+27d+81e+243/+ete. 

(3).  jy=6(r-3)H-c(r2-32)+d(r3-33)+e(r4-34)+/(rs— 3*) 
+0(r6— -36)  etc.+a+36+9c  +  27d+81e+243/+729sr,  etc. 

(4).     N=b(r— 3)+c(r2— 32)+d(r3_ 33)+e  (r4-—  34)  +/(r5— 35) 

(5).  tf =  1 6  (r— 3)  +  c  (r2— 32)  +  <Z  (r3—  33)  +  e  (r4—  34)  +/ 
(r5— 35)+^(r6— 36)+etc.  +  7c+21<H-77e+238/+  7280+ etc.  j  +a 
+36+  2c+6d+4e+5/+ 1  p+etc. 

Now  the  first  part  of  this  expression  is  exactly  divisible  by 

r 3,  or  7  ;  hence  the  only  remainder  that  can  arise  must  occur 

from  dividing  a+36+2c+6d,  etc.,  by  r— 3,  or  7  ;  that  is,  by 
dividing  by  7  the  sum  of  once  the  first  digit,  three  times  the 
second,  two  times  the  third,  six  times  the  fourth,  four  times 
the  fifth,  Jive  times  the  sixth,  and  so  on  in  the  same  order;  and 
when  this  sum  is  exactly  divisible  by  7,  the  number  is  divisi- 
ble by  7.  By  a  slight  change  in  the  terms  of  the  formula,  the 
theorem  as  stated  in  the  second  form  may  also  be  derived. 

Several  years  after  the  discovery  of  the  law  expressed  in 
Principle  2,  I  learned  that  Prof.  Elliott  had  employed  the  same 
property  as  early  as  1846.  Whether  it  was  known  to  any 
mathematicians  previous  to  this  date,  I  am  not  able  to  ascertain. 

Laws  for  Other  Numbers. — In  a  similar  manner  we  may  find 
a  law  for  the  divisibility  of  numbers  by  13,  17,  etc.     The  law 
26 


402  THE    PHILOSOPHY    OF    ARITHMETIC. 

for  13  may  be  stated  as  follows:  A  number  is  divisible  by  13 
when  once  the  first  term,  minus  the  sum  of  3  times  the  second 
4  times  the  third  and  1  time  the  fourth,  plus  the  sum  of  the 
same  multiples  of  the  next  three  terms,  minus  the  sum  of  the 
same  multiples  of  the  next  three  terms,  etc.,  is  divisible  by  13. 

It  will  be  noticed  that  after  the  first  term,  the  series  of  num- 
bers by  which  we  multiply  is  3,  4,  1,  which  is  easily  remem- 
bered and  readily  applied.  To  illustrate,  take  the  number 
8765432 ;  we  have  2— (3  x  3+4  X  4+1  x  5)+(3  X  6+4 X 1+1 X  8) 
=26,  which  is  divisible  by  13;  and  on  trial  we  find  the  num- 
ber itself  is  also  divisible. 

This  law  is  derived  from  the  more  general  principle  that  any 
number  divided  by  13  will  give  the  same  remainder  as  that  ob- 
tained by  dividing  the  result  arising  from  the  above  multiples 
by  13.  This  principle  may  be  demonstrated  by  taking  any 
number,  as  4981654,  and  analyzing  it  as  in  the  previous  case. 

4=  +1X4 

50=  5X10=        5x(13-3)=        5x13-3x5 

600=        6x100=      6x  (104-4)=      6x104-4x6 

4987654=  \       7000=      7x1000=    7  X  (1001-1)=    7x1001-1x7 

80000=    8x10000=    8x(9997+3)=    8x9997+3x8 

900000=  9x100000=  9  X  (99996+4)=  9x99996+4x9 

4000000  =4x  1000000  =4  X  (999999+1) =4  x  999999+1x4 

Laws  for  the  divisibility  of  numbers  by  17,  19,  23,  etc.,  may 

be  obtained  in  a  similar  manner.     We  present  a  few  of  them 

below,  including  7,  11,  and  13,  already  given. 

n  (     1,  3,  2,  -1,  -3,  -2,   1,  3,  2,  -1,  -3,  -2,  etc. 

7*  '  •  |orl,  3,  2,     6,     4,     5,  1,  3,  2,     6,     4,     5,  etc. 

„         (1,  -1,  1,  -1,  1,  -1,  1,  -1,  1,  -1,  1,  -I,  etc. 
U"  *  *  \otI,  10,  1,  10,  1,  10,  1,  10,  1,  10,  1,  10,  etc. 

1Q  (1,  -3,  -4,  -1,  3,  4,  1,  -3,  -4,  -1,  3,  4,  etc. 

l6'  *  *  jorl,  10,     9,  12,  3,  4,  1,  10,     9,  12,  3,  4,  etc. 

„...{ 

orl,  10,  18,  16,  37,  1,  10,  18,  16,  37,  etc 


1,  -7,  -2,  -3,  4,  6,  -8,  5,  -1,  7,  2,  3,  etc. 

or  1,  10,  15,  14,  4,  6,     9,  5,  16,  7,  2,  3,  etc 

f     1,  10,  18,  16,  -4,  1,  10,  18,  16,  -4,  etc. 

V 


THE    DIVISIBILITY    BY    SEVEN.  403 

7q  f     1,  10,  27,  -22,  -1,  -10,  -27,  22,  1,  10,  27,  etc. 

lo'  *  '  lorl,  10,  27,     51,  72,     63,     46,  22,  1,  10,  27,  etc. 
99.  .  .       1,  10,  1,  10,  1,  10,  1,  10,  1,  10,  1,  10,  etc. 

101         j      1,  10,    -1,  -10,  1,  10,    -1,  -10, 1,  10,    -1,  -10,  etc. 
LVXm  #  *jor  1,  10,  100,    91,  1,  10,  100,    91,  1,  10,  100,    91,  etc. 

The  laws  for  99  and  101,  it  is  seen,  are  very  simple  and 
readily  applied. 

ADDITIONAL    PROPERTIES    OF    ELEVEN. 

1.  When  a  number  is  divisible  by  11,  the  number  formed  by 
reversing  its  digits  (its  reverse)  is  divisible  by  11.  Thus  475893 
and  its  reverse  398574  are  both  divisible  by  11. 

2.  The  sum  of  a  number  consisting  of  an  even  number  of  digits 
and  its  reverse  is  divisible  by  11.  Thus  7562  increased  by  2657 
equals  10219  which  is  divisible  by  11. 

3.  The  difference  of  a  number  consisting  of  an  odd  number  of 
digits  and  its  reverse  is  divisible  by  11.  Thus  87652  minus 
25678  equals  61974,  which  is  divisible  by  11. 

4.  A  number  consisting  of  an  even  number  of  digits  is  divisible 
by  11,  when  the  sum  of  the  digits  taken  two  and  two  is  divisible 
by  11.  Thus,  in  475827,  27  +  58  +  47  =  132  which  is  divisible 
by  11  as  is  also  475827. 

5.  A  number  consisting  of  an  odd  number  of  digits  is  divisible 
by  11,  when  the  sum  of  the  units  digit  plus  the  other  digits  taken 
two  by  two  in  a  reverse  order  is  divisible  by  11.  Thus,  in  68596, 
6  +  95  +  86  =  187,  which,  as  also  68596,  is  divisible  by  11. 

6.  If  a  number  consisting  of  an  even  number  of  digits  is  divisi- 
ble by  11,  the  numbers  formed  by  transferring  one  or  more  digits 
in  regular  order  from  either  extremity  of  the  number  to  the  other, 
are  divisible  by  11.  Thus  397452,  974523,  745239,  452397,  etc., 
and  also  239745,  523974,  452397,  etc.,  are  all  divisible  by  11. 

7.  If  a  number  consisting  of  an  odd  number  of  digits  is  divisible 
by  11,  the  numbers  formed  by  transferring  the  digits,  two  by  two, 
reversed,  from  either  extremity  of  the  number  to  the  other,  are 
divisible  by  II.  Thus  2547963,  4796352,  9635274,  etc.,  and 
also  3625479,  9736254,  etc.,  are  all  divisible  by  11. 


CHAPTER  VII. 
PROPERTIES  OF   THE  NUMBER  NINE. 

THE  number  Nine  possesses  the  most  remarkable  pro- 
perties of  any  of  the  natural  numbers.  Many  of  these 
properties  have  been  known  for  centuries  and  have  excited  much 
interest  among  both  mathematicians  and  ordinary  scholars. 
So  striking  and  peculiar  are  some  of  these  properties  that  the 
number  nine  has  been  called  "the  most  romantic  "  of  all  the 
numbers.  On  account  of  its  relation  to  the  numerical  scale,  if 
we  get  the  factor  9  into  a  number  it  will  cling  to  the  expression 
and  turn  up  in  a  variety  of  ways,  now  in  one  place  and  now 
in  another,  in  a  manner  truly  surprising.  It  reminds  one  of  a 
mountain  streamlet  which  ripples  along  its  pathway,  now  buried 
beneath  the  ground  and  for  awhile  hidden  from  our  sight,  but 
presently  gurgling  to  the  surface  at  the  most  unexpected 
moment.  It  is  no  wonder  that  the  property  has  been  regarded 
as  magical,  and  the  number  been  called  the  "magical  number." 
A  few  of  these  interesting  properties  will  be  here  presented. 

1.  The  first  property  of  this  number  which  attracts  our  at- 
tention is,  that  all  through  the  column  of  "nine  times"  in  the 
multiplication  table,  the  sum  of  the  terms  is  nine  or  a  multiple 
of  nine.  Begin  with  twice  nine,  18;  add  the  digits  together, 
and  1  and  8  are  9.  Three  times  9  are  27 ;  2  and  7  are  9.  So 
it  goes  on  up  to  eleven  times  nine,  which  gives  9.9.  Add  the 
digits;  9  and  9  are  18;  8  and  1  are  nine.  Go  on  in  the 
same  manner  to  any  extent,  and  it  is  impossible  to  get  rid  of 
the  figure  9.  Multiply  326  by  9,  and  we  have  2934,  the  sum 
of  whose  digits  is  18,  the  sum  of  whose  digits  is  9.     Let  the 

(404) 


PROPERTIES   OF   THE   NUMBER   NINE.  405 

number  nine  once  enter  any  calculation  involving  multiplica- 
tion, and  whatever  you  do,  "like  the  body  of  Eugene  Aram's 
victim,"  it  is  sure  to  turn  up  again.  This  curious  property  is 
explained  by  the  principle  of  divisibility  of  numbers  presented 
in  the  previous  chapter.  All  these  numbers  being  divisible  by 
9,  the  sums  of  their  digits  must  be  9,  or  a  multiple  of  9. 

2.  Another  curious  property  of  the  number  nine  is  that  if 
you  take  any  row  of  figures  and  change  their  order  as  you 
please,  the  numbers  thus  obtained,  when  divided  by  9,  leave 
the  same  remainder.  Thus,  42378,  24*783,  82734,  etc.,  when 
divided  by  9  all  give  the  same  remainder,  6.  The  reason  of 
this  is,  that  the  sum  of  the  digits  is  the  same,  in  whatever  order 
they  stand ;  and,  as  previously  shown,  the  remainder  from 
dividing  a  number  by  9,  is  the  same  as  the  remainder  from 
dividing  the  sum  of  its  digits  by  9. 

3.  An  interesting  principle  is  presented  in  the  following 
puzzle,  which,  to  the  uninitiated,  seems  very  singular.  Take 
a  number  consisting  of  two  places,  invert  the  figures,  and  take 
the  difference  between  the  resulting  number  and  the  first 
number,  and  tell  me  one  figure  of  the  remainder  and  I  will 
name  the  other.  The  secret  is  that  the  sum  of  the  two  digits 
of  the  remainder  will  always  equal  9.  Thus  take  14,  invert 
the  terms,  and  we  have  47;  take  the  difference  of  the  two  num- 
bers and  we  have  27,  in  which  we  see  that  the  sum  of  7  and  2 
equals  9.  In  this  case,  suppose  I  had  not  known  what  number 
was  taken ;  if  the  person  had  named  one  digit,  say  2,  I  could 
have  immediately  named  the  other  digit  7,  since  I  know  that 
the  sum  of  the  two  digits  is  always  9. 

The  reason  for  this  is  that  both  numbers,  having  the  same 
digits,  are  multiples  of  9  with  the  same  remainder;  hence 
their  difference  is  an  exact  multiple  of  9,  and  consequently  the 
sum  of  the  two  digits  will  equal  9.  When  the  digits  of  the 
number  are  equal,  the  difference  will  be  0;  and  when  they 
differ  by  unity,  the  difference  will  be  9. 

4.  There  is   another   interesting  puzzle,  based  upon  these 


406  THE    PHILOSOPHY    OF    ARITHMETIC. 

principles,  which  is  very  curious  to  one  who  does  not  see  the 
philosophy  of  it,  and  interesting  to  one  who  does.  You  tell  a 
person  to  write  a  number  of  three  or  more  figures ;  divide  by 
d,  and  name  the  remainder;  erase  one  figure  of  the  number; 
divide  by  9,  and  tell  you  the  remainder;  and  you  will  tell  what 
figure  was  erased. 

This  is  readily  done  when  the  principle  is  understood.  If 
the  second  remainder  is  less  than  the  first,  the  figure  erased  is 
the  difference  between  the  remainders;  but  if  the  second 
remainder  is  greater  than  the  first,  the  figure  erased  equals 
the  difference  of  the  remainders  subtracted  from  9.  The 
reason  for  this  is  that  the  remainder,  after  dividing  a  number 
by  9,  is  the  same  as  the  remainder  after  dividing  the  sum  of 
the  digits  by  9,  and  hence  the  sum  of  the  digits  being  diminished 
by  the  number  erased,  the  remainder  will  also  be  diminished 
by  it.  If  there  is  no  remainder  either  time,  then  the  term 
erased  must  be  either  0  or  9. 

To  illustrate,  suppose  the  number  selected  were  457 ;  divid- 
ing by  9  the  remainder  is  7;  erasing  the  second  term  and 
dividing,  the  remainder  is  2 ;  hence  the  term  erased  is  7  less  2 
or  5.  If  the  number  were  461,  dividing  by  9,  the  remainder 
is  2 ;  erasing  the  second  term  and  dividing,  the  remainder  is 
5;  hence  the  term  erased  must  be  the  difference  between  5 
and  2,  or  3,  subtracted  from  9,  which  is  6. 

5.  The  following  puzzle  also  arises  from  the  principle  of  the 
divisibility  by  9.  Take  any  number,  divide  it  by  9,  and  name 
the  remainder;  multiply  the  number  taken  by  some  number 
which  I  name,  and  divide  the  product  by  9,  and  I  will  name 
the  remainder.  To  tell  the  remainder,  I  multiply  the  first 
remainder  by  the  number  which  I  named  as  a  multiplier,  and 
divide  this  product  by  9.  The  remainder  thus  arising  will 
evidently  be  the  same  as  the  remainder  which  the  person 
obtained. 

6.  If  we  take  any  number  consisting  of  three  consecutive 
digits  and,  by  changing  the  place  of  the  digits,  make  two  other 


PROPERTIES   OF   THE   NUMBER   NINE.  407 

numbers,  the  sum  of  these  three  numbers  will  be  divisible  by 
9.  This  depends  on  the  principle  that  the  sum  of  any  three 
consecutive  digits  is  divisible  by  3;  and  consequently  each 
number,  if  not  an  exact  multiple  of  9,  is  a  multiple  of  9  plus  3,  or 
of  9  plus  a  multiple  of  3 ;  and  therefore  the  sum  of  three  numbers 
is  a  multiple  of  9  plus  three  3's,  and  thus  an  exact  multiple  of 
9.  If  we  permutate  the  digits,  making  five  other  numbers,  the 
sum  of  the  six  numbers  will  be  divisible  by  twice  9  ;  which 
may  also  be  readily  explained. 

7.  From  the  law  of  the  divisibility  by  nine,  several  other 
properties,  especially  interesting  to  the  young  arithmetician, 
may  be  derived.  Among  these  may  be  mentioned  the  follow- 
ing: 1.  If  we  subtract  the  sum  of  the  digits  from  any  number 
the  difference  will  be  exactly  divisible  by  9.  2.  If  we  take 
two  numbers  in  which  the  sums  of  the  digits  are  the  same,  the 
difference  of  the  two  numbers  will  be  divisible  by  9.  3.  Ar- 
range the  terms  of  any  number  in  whatever  order  we  choose, 
and  divide  by  9,  and  the  remainder  in  each  case  is  the  same. 
Such  properties  as  these  must  have  seemed  exceedingly  curious 
to  the  early  arithmeticians,  and  fully  entitle  the  number  nine 
to  be  regarded  as  a  magical  number.  All  of  these  properties, 
it  is  proper  to  remark,  would  have  belonged  to  the  numb'.j* 
eleven,  if  our  scale  had  been  duodecimal  instead  of  decimal. 


CHAPTER  VIII. 

INTERESTING    PROPERTIES    OF    NUMBERS. 

THERE  are  many  interesting  properties  of  numbers  besides 
those  already  presented,  a  few  of  which  will  be  given  in  this 
chapter.  Some  of  these  will  admit  of  demonstration,  but  for 
others  no  demonstration  has  yet  been  found.  Those  that  have 
not  been  proved  can  be  shown  to  be  true  in  so  many  special  cases 
that  it  can  reasonably  be  inferred  that  they  are  true  in  all  cases. 

1.  The  sum  of  the  first  n  odd  numbers  is  equal  to  "of-     Thus 

1  +  3  =  22;   1  +  3  +  5  =  32;   1  +  3  +  5  +  7  =  42;    1  +  3  +  5  + 
7  +  9  =  52- 

2.  The  sum  of  the  first  n  even  numbers  is  equal  to  n2  +  n.     Thus 

2  +  4  =  22 +  2;    2  +  4  +  6  =  32 +  3;    2  +  4+6  +  8  =  4' +  4; 
2  +  4  +  6  +  8  +  10  =  52  +  5  ;  etc. 

3.  Every  cubic  number,  if,  is  the  sum  of  XL  odd  numbers.  Thus 
2'  =  3  +  5;  33  =  7  +  9  +  ll;  43  =  13  +  15  +  17  +  19  ;  53  = 
21  +  23  +  25  +  27  +  29  ;  63  =  31  +  33  +  35  +  37.  +  39  +  41 ;  etc. 

4.  The  difference  between  the  squares  of  any  two  consecutive 
numbers  is  the  sum  of  the  two  numbers.  Thus  32  —  22  =  3  +  2  ; 
42-32  =  44.3.  52  —  43  =  5  +  4;  62  —  52  =  6  +  5  ;  etc.  This 
is  readily  proved,  as  (?i  +  l)2  —  n2  =  2?i  +  1  =  (n  +  1)  +  n. 

5.  The  sum  of  two  squares  multiplied  by  the  sum  of  two  squares 
is  always  equal  to  the  sum  of  two  squares.  Thus  (l2  +  22)  (22  + 
32)^65  =  l2  +  82;  (l2  +  42)(32  +  42)  =  125  =  52  +  102 ;  (12  + 
22)  (52  +  62)  =  305  ==  42  +  172;  (22  +  32)  (4s  +  52)  =  72  +  222;  etc. 

G.  TJie  sum  of  the  cubes  of  the  natural  numbers  equals  the  square 
of  the  sum  of  these  numbers.  Thus  l3  +  23  +  33  =  (1  +  2  +  3)2 ; 
I8  +  23  +  33  +  43  =  (1  +  2  +  3  +  4)2- 

(408) 


INTERESTING    PROPERTIES    OF    NUMBERS.  409 

7.  The  difference  between  a  number  and  its  cube  is  the  product 
of  three  consecutive  numbers  the  second  of  which  is  the  given  number. 
Thus  33  —  3  =  2X3X4.  This  is  proved  as  follows  :  nz  —  n  = 
n(n2—\)  —  n(n  +  1)  (n  —  1)  and  n  —  1,  n  and  n  +  1. 

8.  Every  prime  number  (in  general  in  more  than  one  way)  is 
the  sum  of  four  square  numbers.     Thus  1  =  (^)2  +  (i)2  +  (^)2  + 

(i)2;    2  =  (|)'  +  (f)2  +  (i)2  +  (f)2;    3  =  (i)2  +  (i)2  +  (i)2  + 
(f)2;5  =  (i)2  +  (i)2+(i)2  +  (f)2;7  =  l2  +  l2+l2  +  22;.    .    . 
13  =  22  +  22  +  22  +  l2;  .    .    .  23  =  32  +  32  +  22  +  l2;  .    .    .31  = 
22  +  32  +  32  +  32 ;  37  =  2*  +  22  +  22  +  52 ;  etc. 

9.  Any  prime  number  which  when  divided  by  4  leaves  a  re- 
mainder of  unity,  is  the  sum  of  two  square  numbers.  Thus  13  = 
22  +  32 ;  17  =  1  +  42 ;  29  =  22  +  52 ;  37  =  l2  +  G2 ;  41  =  43  +  5' ; 
53  =  42  +  72 .  59  =  52  +  G2 ;  89  =  o2  +  82 ;  233  =  8'  +  132 ;  etc. 

10.  Every  even  number  is  the  sum  of  two  prime  numbers.  This 
is  known  as  Goldbach's  theorem.  Thus  4  =  1  +  3,  6=1  +  5; 
8  =  3  +  5;  10  =  3  + 7;  12  =  5  + 7;  14  =  3 +  11;  16  =  3+ 13; 
18  =  1  +  17,  or  5 +  13;  etc. 

11.  Every  square  number  is  the  sum  of  either  two  or  of  three 
prime  numbers.  Thus  4=1  +  3;  9  =  2  +  7;  16  =  5+11; 
25  =  1  +  5+19;  36  =  17  +  19;  49  =  2  +  47;  172  =  3  +  5  + 
281;  232  =  1 +  5  +  523;  242  =  5  +  571;  252=l  +  5  +  619; 
482  =  5  +  2297  ;  etc. 

12.  The  sum  of  the  same  powers  (above  the  second)  of  two  num- 
bers cannot  equal  a  perfect  power.  This  is  known  as  Fermat's 
theorem.     It  has  never  been  demonstrated. 

13.  The  consecutive  integers  8  and  9  are  exact  powers;  are  there 
any  other  consecutive  integers  which  are  exact  powers  f  This 
problem  is  ascribed  to  Catalan.  So  far  as  known  it  has  never 
received  a  demonstration. 

14.  Five  persons  can  be  seated  in  six  different  ways  around  a 
table  in  such  a  manner  that  any  one  person  is  seated  only  once  be- 
tween the  same  two  persons.  The  manner  of  seating  is  as  follows : 
l_2-3-4-5;  1-2-4-5-3;  1-3-2-5-4;  1-3-4-2-5;  1-4-2-3-5; 
l_4_3-5-2. 


410  THE   PHILOSOPHY    OF   ARITHMETIC. 

15.  Six  persons  can  be  seated  in  ten  different  ways  around  a 
table  in  such  a  manner  that  any  one  person  is  seated  only  once  be- 
tween the  same  two  persons.     Show  the  manner  of  seating. 

1 6.  Seven  persons  may  be  seated  in  fifteen  different  ways  around 
a  table  in  such  a  manner  that  any  one  person  is  seated  only  once  be- 
tween the  same  two  persons.  Show  the  ways  in  which  they  may 
be  seated. 

This  last  problem  will  afford  interesting  pastime  for  any  one 
who  attempts  its  solution.  I  have  known  many  to  fail  in  working 
it,  and  only  two  or  three  to  succeed.  The  problem  is  possible, 
however,  and  I  have  several  different  solutions  of  it. 

17.  The  nine  digits  {including  the  cipher)  may  be  combined  in 
many  ways  so  as  to  make  100.  The  following  are  some  of  the 
methods  of  combination : 

(1)        (2)  (3)  (4)  (5)  (6)        (7)  (8)  (9) 

79||       67H       87ff       92f£       89ft      89|J      96r8ft      92T5ft      76ft 
20i         82f         12f  7ft       10f        10f  3|  7*  23ft 

100  100         100         100         100        100        100  100  100 

(10)        (11)  (12)  (13)        (14)  (15)      (16)       (17)      (18) 

1W       Wit       97t¥o        97|§       97ft\       97ff      95f|      79ft      79* 
21  f  2ft  2f  2ft  2|  2ft        41        14f        15f 

6  4 


100  100         100  100  100  100        100        100        100 

The  author  has  thirty-three  solutions  and  there  may  be  many 
others. 


Part  IV. 

FRACTIONS. 
SECTION  I. 

COMMON   FRACTIONS. 


I.  Nature  op  Fractions. 


II.  Classes  of  Common  Fractions, 


III.  Treatment  op  Common  Fractions. 


IY.  Continued  Fractions. 


CHAPTER  L 

NATURE   OF   FRACTIONS. 

THE  Unit  is  the  fundamental  idea  of  arithmetic.  From  it 
arise  two  great  classes  of  numbers — Integers  and  Frac- 
tions. Integers  have  their  origin  in  the  multiplication  of  the 
Unit;  Fractions  arise  from  the  division  of  the  Unit.  One  is 
the  result  of  an  immediate  synthesis;  the  other,  of  a  primary 
analysis.  Fractions  have  their  origin  in  the  analysis  of  the 
Unit,  as  integers  arise  from  the  synthesis  of  units. 

When  the  Unit  is  divided  into  equal  parts,  each  part  is  seen 
to  bear  a  certain  relation  to  the  Unit,  and  these  parts  may  be 
collected  together  and  numbered.  This  complex  process  of  di 
vision,  relation,  and  collection,  gives  us  a  fraction.  The  con 
ception  of  a  fraction,  therefore,  involves  three  things : — 1st,  a 
division  of  the  unit;  2d,  a  comparison  of  the  part  with  the 
unit ;  3d,  a  collection  of  the  equal  parts  considered.  When  a 
unit  is  divided  into  a  number  of  equal  parts,  the  comparison 
of  the  part  with  the  unit  gives  the  fractional  idea,  and  the  col- 
lection of  the  parts  gives  the  fraction  itself.  Herein  is  clearly 
seen  the  distinction  between  an  integer  and  a  fraction.  The 
former  is  an  immediate  synthesis;  the  latter  involves  a  process 
of  division,  an  idea  of  relation,  and  a  synthesis  of  the  parts. 
A  fraction  is,  therefore,  a  triune  product — a  result  of  analysis, 
comparison,  and  synthesis. 

Fractions,  as  has  been  stated,  have  their  origin  in  a  division 
of  the  Unit ;  they  may  also  be  derived  from  the  comparison  of 
numbers.  Thus  the  comparison  of  one  with  two,  or  of  two  with 
four,  may  give  the  idea  of  one-half  ;  and  in  a  similar  manner 

(413)  " 


414  THE    PHILOSOPHY    OF    ARITHMETIC. 

other  fractions  may  be  obtained.  This,  however,  is  a  possible 
rather  than  the  actual  origin ;  fractions  really  originated  in  the 
division  of  the  Unit. 

When  the  Unit  is  divided  into  equal  parts,  these  parts  are 
collected  and  numbered  as  individual  things  ;  they  may,  there- 
fore, be  regarded  as  a  special  kind  of  units.  To  distinguish 
them  from  the  Unit  already  considered,  we  call  them  fraction at 
units.  This  gives  us  two  classes  of  units,  integral  units  and 
fractional  units.  The  integral  unit  is  known  as  the  Unit ; 
when  fractional  units  are  meant  we  use  the  distinguishing 
term  fractional.  The  definite  conception  of  an  integer  requires 
a  clear  idea  of  the  Unit ;  the  definite  conception  of  a  fraction 
requires  a  clear  idea  both  of  the  integral  and  the  fractional  unit. 
The  character  of  the  thing  divided,  and  the  nature  of  the  divis- 
ion, must  be  kept  clearly  before  the  mind,  in  order  to  obtain  a 
distinct  conception  of  the  fraction.  From  this  brief  statement 
of  the  nature  of  the  fraction  we  are  prepared  to  define  it. 

Definition. — A  fraction  is  a  number  of  the  equal  parts  of  a 
Unit.  This  definition  is  an  immediate  inference  from  the  con- 
ception of  a  fraction  above  presented.  We  divide  the  Unit 
into  equal  parts,  and  then  take  a  number  of  these  equal  parts, 
and  this  is  the  fraction.  A  definition  quite  similar  to  this  is, 
a  fraction  is  one  or  more  of  the  equal  parts  of  a  unit.  This 
is  not  incorrect,  though  it  is  preferred  to  use  the  word  "  num- 
ber "  for  "  one  or  more."  It  is  believed  that  the  idea  is  thus 
expressed  in  the  most  concise  and  elegant  form,  and  that  it  will 
meet  the  approval  of  mathematicians. 

Several  other  definitions  of  a  fraction  have  been  presented 
by  different  authors,  some  of  which  are  correct,  while  others  are 
liable  to  serious  objections.  One  writer  says,  "A  fraction  is  a 
part  of  a  unit."  This  is  only  part  of  the  truth,  for  a  fraction 
may  be  not  only  one  part  but  several  parts  of  a  unit.  Another 
writer  says,  "A  fraction  is  an  expression  for  one  or  more  of 
the  equal  parts  of  a  unit."  In  this  definition  the  expression, 
the  written  or  printed  symbols,  is  made  the  fraction,  which  is 


NATURE   OF   FRACTIONS.  415 

evidently  incorrect,  as  we  have  fractions  previous  to  and  inde- 
pendent of  the  expression  of  them.  The  expressions  are  not 
subjects  of  mathematical  calculation,  and  hence  they  cannot  be 
fractions.  The  same  distinction  holds  between  a  fraction  and 
its  expression,  as  between  a  number  and  its  expression.  Thus 
we  have  the  number  four  and  the  figure  4 ;  so  we  have  the 
fraction  three-fourths,  and  the  expression  f,  as  two  distinct 

things. 

Another  definition  of  a  fraction  is  that  it  is  an  "unexecuted 
division."  Says  one  writer,  "  A  fraction  is  nothing  more  nor 
less  than  an  unexecuted  division."  Says  another,  "  A  fraction 
may  be  regarded  as  an  expression  of  an  unexecuted  division." 
This  conception  of  a  fraction  is  incorrect,  as  the  idea  of  a  frac- 
tion, and  the  idea  of  the  division  of  one  number  by  another, 
are  entirely  distinct.  The  fraction  i  (4  fifths),  means  four  of 
the  equal  parts  which  are  obtained  by  dividing  a  unit  into  five 
equal  parts.  The  division  of  4  by  5  will  give  the  expression 
*,  but  the  idea  of  4  divided  by  5  is  entirely  distinct  from  the 
fractional  idea ;  and  hence  the  assertion,  that  a  fraction  is  nothing 
more  nor  less  than  an  unexecuted  division,  is  absurd. 

A  fraction  has  also  been  defined  to  be  the  relation  of  a  part 
of  anything  to  the  whole.  This  was  the  idea  of  Sir  Isaac 
Newton,  and  is  correct,  though  it  is  rather  too  abstract  for  a 
popular  definition.  Another  form  of  stating  the  same  idea  is 
that  "  a  fraction  is  that  definite  part  which  a  portion  is  of  the 
whole."  Thus,  if  we  divide  an  apple  into  two  equal  por- 
tions, one  of  these  is  one-half  of  the  whole,  and  this  definite 
part,  one-half,  is  the  fraction.  This  form  of  statement  is  not 
incorrect,  though,  like  Newton's,  it  is  too  abstract  for  a  popular 
definition. 

Notation.— A  fraction  being  a  number  of  equal  parts  of  a 
unit,  it  is  natural  that,  in  the  notation  of  a  fraction,  we  should 
indicate  the  number  of  parts  used,  by  a  figure.  It  would  also, 
at  first  thought,  seem  natural  to  represent  the  name  of  the  frac- 
tional unit  by  the  words,  half  third,  etc.,  as  2  thirds,  3  fourths, 


416  THE   PHILOSOPHY    OF   ARITHMETIC. 

etc.;  or  by  their  abbreviations,  as  2-3ds,  3-4 ths,  etc.  The  let 
ters  would  be  finally  omitted  altogether,  and  the  expressions 
become  2-3,  3-4,  etc.  This  probably  was  the  primary  form,  as 
is  indicated  by  the  expressions,  2-3  for  2  thirds,  3-4  for  3 
fourths,  which  we  meet  in  some  of  the  older  books. 

It  has  been  found  more  convenient,  however,  not  to  express 
directly  the  name  of  the  part,  but  rather  to  represent  the  num- 
ber of  parts  into  which  the  unit  is  divided,  from  which  the 
name  of  the  part  is  inferred.  This  might  have  been  done  by 
writing  one  figure  after  another,  2-3,  the  3  denoting  the  number 
of  equal  parts  of  the  unit',  and  the  2  the  number  of  parts  con- 
sidered. In  practice  it  has  been  agreed,  however,  to  write  the 
figure  denoting  the  number  of  parts  into  which  the  unit  is 
divided,  under  the  other,  separating  them  by  a  line,  as  in  divi- 
sion. The  number  expressed  by  the  figure  below  the  line  is 
called  the  denominator  of  the  fraction,  the  number  expressed 
by  the  figure  above  the  line  is  called  the  numerator  of  the 
fraction.  The  primary  object  of  the  figure  below  the  line  is  not 
to  name  the  fractional  unit,  but  to  denote  the  number  of  equal 
parts  of  a  unit ;  from  this  the  name  of  the  fractional  unit  is  in- 
ferred. Primarily,  then,  in  our  present  notation,  the  denomi- 
nator of  the  fraction  is  not  the  denomination  of  the  fraction, 
though  from  the  denominator  the  denomination  is  inferred. 
The  denominator  thus  serves  the  double  object  of  showing 
directly  the  number  of  equal  parts  into  which  the  unit  is  divided, 
and,  indirectly,  the  name  or  denomination  of  the  fraction.  This 
distinction  should  be  carefully  noted. 

In  integers  we  have  one  word  to  indicate  the  thing  itself, 
and  another  to  indicate  the  expression  of  it.  Thus,  number 
means  the  how  many,  or  thing  itself;  and  figure,  the  expres- 
sion of  it;  the  thing  and  its  symbol  being  distinguished  by  in- 
dependent names.  In  fractions  there  are  no  such  terms  to  distin- 
guish the  expression  of  a  fraction  from  the  fraction  itself.  "We 
are  therefore  obliged  to  use  the  same  word  fraction  to  designate 
both.     This  we  are  authorized  to  do  by  a  figure  of  rhetoric 


NATURE   OF  FRACTIONS.  417 

called  Metonymy,  in  which  the  name  of  an  object  is  sometimes 
given  to  the  symbol,  or  expression  of  the  object.  It  is  conse- 
quently allowable  to  use  the  word  fraction  when  we  mean  the 
expression  of  a  fraction,  though  this  frequently  occasions  con- 
fusion and  calls  for  particular  care  on  the  part  of  the  teacher 
to  prevent  it.  We  are  sometimes  obliged  to  make  the 
same  dual  use  of  the  terms  numerator  and  denominator,  but 
should  always  do  so  with  extreme  caution  to  avoid  confusion. 

The  expression  of  a  fraction  in  its  relation  to  the  fraction 
itself,  is  seen,  when  analyzed,  to  be  a  more  complicated  thing 
than  at  first  appears.  To  illustrate;  first,  we  have  the  fraction 
itself,  as  so  many  parts  of  a  unit ;  then  we  have  the  two  figures 
to  represent  the  fraction;  and  then  we  have  the  numbers,  which 
these  two  figures  denote ;  all  of  which  should  be  carefully  dis- 
tinguished, if  we  would  have  a  clear  idea  of  the  relation  of  a 
fraction  to  its  notation.  If  we  begin  with  the  unit  and  com- 
pare it  with  the  fraction  as  expressed,  the  matter  becomes  still 
more  complicated.  Thus,  first  we  have  the  Unit;  then  the 
equal  parts  into  which  the  Unit  is  divided ;  then  the  relation 
of  these  parts  to  the  Unit ;  then  the  expression  for  a  number 
of  these  parts,  consisting  of  two  figures;  and  then  the  numbers 
which  these  figures  denote.  It  is  therefore  not  entirely  sur- 
prising that  writers  should  have  been  careless  and  confused  in 
their  use  of  the  terms  relating  to  fractions. 

History Before  proceeding  to  the  classification  and  treat- 
ment of  Fractions,  attention  is  called  to  a  few  points  concerning 
their  origin  and  history.  The  treatment  of  fractions  by  Ahmes 
is  shown  in  the  chapter  en  the  origin  of  our  system  of  arith- 
metic. In  the  Lilawati,  fractions  are  denoted  by  writing  the 
numerator  above  the  denominator,  without  any  line  between 
them.  The  introduction  of  the  line  of  separation  is  due  to  the 
Arabs ;  and  it  is  found  in  their  earliest  manuscripts  on  arith- 
metic. To  denote  a  fraction  of  a  fraction,  as  §  of  |,  the  two 
fractions  are  written  consecutively,  without  any  symbol  between 
them.  To  represent  a  number  increased  by  a  fraction,  the 
27 


STATEMENT. 

1 

12     3     11 

1 

1 

2     3     4     5  16 

4 

418  THE   PHILOSOPHY   OF   ARITHMETIC. 

fraction  is  written  beneath  the  number ;    and  when  the  fraction 
is  to  be  subtracted  from  the  number  a  dot  is  prefixed  to  it;  thus, 

2  3 

2£  is  denoted  by  x  and  3— ±  by#1 

In  other  cases,  their  notation  is  not  intelligible  without  ver- 
bal explanation,  and  the  same  is  true  of  the  Arabs  and  earlier 
European  writers,  who  were  singularly  deficient  in  artifices  of 
notation.  In  the  solution  of  a  problem 
in  the  Lilawati,  in  which  "  the  fourth 
of  a  sixteenth  of  the  fifth  of  three 
quarters  of  two-thirds  of  a  moiety"  is 
required,  the  work  is  written  as  indicated  in  the  margin ; 
which  gives  y^f  or  T^v.  In  solving  the  problem,  "Tell  me, 
dear  woman,  quickly,  how  much  a  fifth, 
a   quarter,  a   third,  a   half  and  a  sixth  statement. 

11111     29 
make  when  added  together,"    the  work        54326     20 

appears  in  the  Lilawati  as  indicated  in 

the  margin.      In  solving  the  problem,  "Tell  me  what  is  the 
residue  of  three,  subtracting  these  frac- 
tions ;"  they  expressed  the  work  as  in-  statement. 
dicated,  which  it  is  apparent  could  not       154326      20 
be  understood  without  an  explanation. 

The  Lilawati  contains  four  rules  for  the  reduction  and  as- 
similation of  fractions,  as  well  as  the  application  of  their  eight 
fundamental  rules  of  arithmetic  to  them.  These  rules  are  clear 
and  simple,  and  differ  very  little  from  those  used  in  modern 
practice.  That  the  author  regarded  fractions  as  somewhat 
difficult,  is  apparent  from  the  following  problem :  "  Tell  me  the 
result  of  dividing  five  by  two  and  a  third,  and  a  sixth  by  a 
third,  if  thy  understanding,  sharpened  into  confidence,  be  com- 
petent to  the  division  of  fractions." 

The  notation  of  compound  fractions  varied  with  different 
authors;  thus  with  Lucas  di  Borgo  §  of  f,  or  fxf,  va 

was  represented  as  in  the  margin,  where  va  denotes     I £ 

via,   or   times.     Stifel   denoted  three-fourths  of  two- 


NATURE     OP     FRACTIONS.  419 

thirds  of  one-seventh  by  writing  the  fractions  nearly  f 

under   one  another  as  in  the  margin ;    and  the   same        -§ 
operation  was  indicated  by  Gemma  Frisius  thus :  ^ 

fin* 

a  notation  simple  and  convenient. 

In  the  writings  of  Lucas  di  Borgo,  when  two  fractions  are 
to  be  added   together  or   subtracted    one   from   another,  the 
operations  to  be  performed  are  indicated  as  follows : 
8     9 

12 
where  those  quantities  are  to  be  multiplied  together  which  are 
connected  by  the  lines.  There  seems  to  be  very  little  difference 
between  the  operations  in  fractions  in  ancient  and  modern  text- 
books. In  the  works  of  Di  Borgo  and  Tartaglia,  the  number  of 
cases  and  their  subdivisions  are  unnecessarily  multiplied,  and 
the  reader  is  frequently  more  perplexed  than  instructed  by  the 
minuteness  of  their  explanations.  It  may  be  remarked  that  the 
early  writers  seem  to  have  been  extremely  embarrassed  by  the 
usage  and  meaning  of  the  term  multiplication  in  the  case  of 
fractions,  where  the  product  is  less  than  the  multiplicand;  and 
some  of  their  methods  of  explaining  the  seeming  inconsistency 
are  curious  and  ingenious. 


CHAPTER  II. 

CLASSES   OF   COMMON  FRACTIONS. 

FRACTIONS  are  divided  into  two  general  classes — Com- 
mon and  Decimal.  A  Common  Fraction  is  a  number  of 
equal  parts  of  a  unit,  without  any  restriction  as  to  the  size  of 
those  parts.  A  Decimal  Fraction  is  a  number  of  the  decimal 
divisions  of  a  unit ;  that  is,  a  number  of  tenths,  hundredths, 
etc. 

This  distinction  of  fractions  originated  in  a  difference  in  the 
notation,  rather  than  in  any  essential  difference  in  the  fractions 
themselves.  It  was  seen  that  the  decimal  scale  of  notation, 
when  extended  to  the  right  of  the  units  place,  was  capable  of 
expressing  tenths,  hundredths,  etc.,  and  that  there  would  be  a 
great  advantage  in  such  an  expression  of  them ;  and  thus  the 
decimal  fraction  came  to  be  regarded  and  treated  as  a  distinct 
class.     A  brief  discussion  of  each  will  be  given. 

Common  Fractions  are  variously  classified,  according  to  dif- 
ferent considerations.  The  primary  division  is  that  based  upon 
their  relative  value  compared  with  the  Unit.  Classifying  them 
in  reference  to  this  relation,  we  have  Proper  Fractions  and 
Improper  Fractions.  A  Proper  Fraction  is  one  whose  value 
is  less  than  a  unit;  that  is,  one  which  is  properly  a  fraction  ac- 
cording to  the  primary  conception  of  a  fraction.  An  Improper 
Fraction  is  one  which  is  equal  to  or  greater  than  a  unit ;  that 
is,  one  which  is  not  properly  a  fraction  in  the  primary  meaning 
of  the  term. 

Another  division  of  common  fractions  arises  from  the  idea 
of  dividing  a  fraction  into  equal  parts.     A  fraction  originated 

(420) 


CLASSES   OP   COMMON   FRACTIONS.  421 

in  the  division  of  the  Unit  into  equal  parts;  now,  if  we  ex- 
tend this  idea  to  obtaining  a  number  of  equal  parts  of  a  fraction, 
we  get  what  is  called  a  Compound  Fraction.  The  Compound 
Fraction,  it  is  thus  seen,  originated  in  the  extension  of  the 
primary  idea  of  division,  which  gave  rise  to  the  simple  fraction 
This  idea  of  a  compound  fraction  leads  to  the  division  of  frac- 
tions into  two  classes — Simple  Fractions  and  Compound  Frac- 
tions. A  Compound  Fraction  is  technically  denned  as  a  fraction 
of  a  fraction. 

If  we  extend  the  fractional  idea  a  little  further,  and  suppose 
the  numerator,  or  denominator,  or  both,  to  become  fractional,  we 
have  what  arithmeticians  call  a  Complex  Fraction.  The 
Complex  Fraction  may  be  denned  as  a  fraction  whose  numera- 
tor, or  denominator,  or  both,  are  fractional.  Whether  the  com- 
plex fraction  agrees  with  the  definition  of  a  fraction,  or  with 
the  functions  ascribed  to  the  numerator  or  the  denominator  of 
a  fraction,  is  a  point  which  will  be  considered  a  little  further 
on ;  but  its  origin  was  a  natural  outgrowth  of  the  principle  of 
pushing  a  notation  to  its  limits.  It  should  be  noticed  that  the 
complex  fraction  may  also  have  originated  in  the  expression 
of  the  division  of  one  fraction  by  another  by  writing  the  divisor 
under  the  dividend  with  a  line  between  them;  but  the  proba- 
bilities are  that  it  originated  as  first  indicated,  by  an  extension 
of  the  fractional  idea. 

Fractions,  therefore,  are  divided  with  regard  to  their  value,  as 
compared  with  the  Unit,  into  Proper  and  Improper  Fractions; 
with  regard  to  their  form,  into  Simple,  Compound,  and  Com- 
plex. There  is  also  another  form  of  expressing  fractional  rela- 
tions, so  closely  connected  with  the  common  fraction  that  it 
may  be  embraced  under  the  same  general  head.  I  refer  to  the 
Continued  Fraction,  which  will  be  treated  with  the  general 
subject  of  common  fractions. 

Improper  Fractions. — According  to  the  primary  idea,  a 
fraction  is  regarded  as  a  part  of  a  unit,  and  hence  as  less  than 
a  unit.     But  since  we  can  speak  of  any  number  of  fractional 


4:22  THE   PHILOSOPHY    OF    ARITHMETIC. 

units  as  we  do  of  integral  units,  there  arises  a  fractional 
expression  whose  value  is  greater  than  a  unit.  Thus  we  may 
speak  of  5  fourths,  1  fourths,  etc.,  although  in  a  unit  there  are 
only  4  fourths.  These  we  call  improper  fractions;  that  is 
they  are  improperly  fractions  from  the  primary  idea  of  a 
fraction.  The  improper  fraction  presents  several  points  of 
difficulty  and  interest,  which  will  be  briefly  considered. 

Take  the  expression  $£;  is  this  strictly  a  fraction?  That  it 
is  properly  a  fraction,  appears  from  the  definition  of  a  fraction 
and  from  the  discussion  just  given.  How,  then,  shall  it  be 
read?  If  we  read  it  "f  of  a  dollar,"  some  one  will  object, 
that  there  are  only  four  fourths  in  a  dollar,  and  hence  you 
cannot  speak  of  five  fourths  of  a  dollar.  If  it  be  read  "  £  dollars," 
we  will  object,  since  there  are  not  enough  to  make  dollars,  the 
plural  meaning  two  or  more.  But,  says  some  one,  the  gram- 
mars tell  us  that  "the  plural  means  more  than  one,"  and  since 
$|  is  more  than  one,  we  may  use  the  plural  form  and  say  "£ 
dollars."  This,  we  reply,  is  a  mere  quibble,  as  the  grammar- 
ians contemplate  only  integers  when  they  say  "more  than 
one,"  and  really  mean  "two  or  more."  The  reading  "f  dol- 
lars" is,  therefore,  not  strictly  correct. 

How,  then,  should  it  be  read?  I  think  the  correct  reading 
is  "J  of  a  dollar."  We  mean  by  it  five  of  such  parts  as  are 
obtained  by  dividing  a  dollar  into  four  equal  parts.  It  is  true 
there  are  not  five  fourths  in  one  dollar,  and  the  reading  does 
not  assume  that  there  are.  No  one  will  object  to  saying  J  of 
100  cents  equals  125  cents,  which  is  equivalent  to  saying  £  of 
a  dollar  equals  a  dollar  and  a  quarter.  The  fractional  units 
are  fourths  of  a  dollar,  and  the  number  of  fractional  units  is 
five;  hence  the  fraction  is  "five-fourths  of  a  dollar."  It  is  an 
improper  fraction — improperly  a  fraction  from  the  primary 
idea  of  a  fraction — and  in  the  name  "improper  fraction"  we 
apparently  enter  a  little  protest  against  the  absolute  correctness 
of  the  reading  in  view  of  the  primary  idea  of  the  fraction.  If 
we  have  $f  or  $-^,  we  can  then  say  f  dollars  or  ^  dollars, 


CLASSES   OP   COMMON   FRACTIONS.  423 

since  we  then  have  "two  or  more."  This  discussion  seems  to 
have  been  called  for  from  the  fact  that  the  question  is  often 
raised  and  debated  as  to  what  is  the  correct  reading  of  the 
improper  fraction. 

Complex  Fraction. — According  to  the  strictest  definition  of 
a  fraction,  the  complex  fraction  is  an  impossibility.  This  is 
rendered  evident  from  a  consideration  of  the  functions  ascribed 
to  the  denominator  by  the  definition.  The  denominator  shows 
the  number  of  equal  parts  into  which   the  unit  is  divided  ; 

hence,  in  the  complex  fraction  ~,  the  denominator,  f ,  denotes 

% 
that  the  unit  is  divided  into  f  equal  parts.  This  is  an  impos- 
sibility, as  may  be  seen  at  least  in  two  ways.  First,  we  can 
divide  a  unit  into  three  or  two  equal  parts,  but  not  into  one 
part,  since  there  will  be  no  division;  and  if  we  cannot  divide 
it  into  one  equal  part,  it  is  evident  that  we  cannot  divide  it 
into  less  than  one  equal  part.  Secondly,  if  any  one  doubts 
the  conclusion  from  this  reasoning,  let  him  take  an  apple  and 
endeavor  to  divide  it  into  f  equal  parts.  The  effort  I  have 
sometimes  known  to  be  in  a  high  degree  amusing,  and  always 
conclusive  of  the  correctness  of  the  position  assumed  above. 

A  somewhat  plausible  argument  in  favor  of  the  correctness 
of  the  complex  fraction  is  the  following :  In  the  algebraic  frac- 
tion 7,  the  numerator  and  denominator  are  general  expressions, 
b 

and  hence  may  represent  fractions  as  well  as  integers.  If  then 
6=  J  we  shall  have  a  complex  fraction.  This  method  of  reason- 
ing is  too  general  for  arithmetic ;  even  in  algebra  it  would  prove 

CLOT        G 

that  clearing  the  equation,  -j-=-i,  of  fractions,  does  not  clear  it 

of  fractions,  since  in  adx=bc,  each  term  may  be  a  fraction. 

The  expression  -  means  a  divided  by  b,  and  is  a  fraction  only 

so  far  as  it  coincides  with  our  arithmetical  idea  of  a  fraction 
We  conclude,  therefore,  that  strictly  speaking,  the  complex 
fraction  is  an  impossibility.  It  is  merely  a  convenient  expres- 
sion that  one  fraction  is  to  be  divided  by  another. 


424  THE    PHILOSOPHY   OF    ARITHMETIC. 

Should  the  idea  and  expression  of  a  complex  fraction,  there- 
fore, be  discarded  from  arithmetic  ?  This  does  not  follow,  and 
is  not  recommended.  It  is  a  convenient  form  of  expressing  the 
division  of  one  fraction  by  another,  and  may  thus  be  retained. 
Those  who  use  it,  however,  should  understand  that  it  is  not 
strictly  a  fraction,  according  to  the  primary  idea  of  a  fraction, 
but  a  representation  of  the  division  of  a  fraction  by  a  fraction, 
or  of  a  whole  number  and  a  fraction  when  only  one  term  is 
fractional. 

Is  a  Fraction  a  Number  ?  It  has  been  stated  by  some  writers, 
and  seems  frequently  to  be  the  idea  of  pupils,  that  a  fraction  is 
not  a  number.  This,  however,  is  a  mistake,  as  will  appear 
from  a  slight  consideration  of  the  matter.  Newton's  definition 
of  number  provides  for  the  fractional  number  when  the  object 
measured  is  a  definite  part  of  the  measure ;  it  consequently  ap- 
pears that  the  fraction  is  a  number,  if  we  accept  his  definition 
as  correct.  The  definition,  "A  Fraction  is  a  number  of  equal 
parts  of  unity,"  also  makes  it  clear  that  a  fraction  is  a  num- 
ber. Again,  if  it  is  not  a  number,  what  kind  of  a  quantity  is 
it ;  and  why  should  it  be  treated  in  arithmetic,  the  science  of 
numbers  ?  Five  inches  is  certainly  a  number ;  hence  its  equiv- 
alent, five-twelfths  of  a  foot,  is  also  a  number.  Numbers  are 
of  two  classes,  integers  and  fractions  ;  and  fractions  are  num- 
bers, as  much  so  as  integers.  The  fractional  number,  it  will  be 
noticed,  involves  two  ideas — first,  the  integral  unit;  and  second, 
the  fractional  unit.  In  an  integer  we  have  the  idea  of  a  num- 
ber of  units ;  in  the  fraction  we  have,  not  only  an  idea  of  a 
number  of  units,  but  also  the  relation  of  the  fractional  unit  to 
the  integral  unit. 

Is  a  Fraction  a  Denominate  Number  ?  It  has  been  affirmed 
by  some  authors  that  "fractions  are  a  species  of  denominate 
numbers."  This,  however,  is  true  only  in  a  very  limited  or 
partial  sense.  Three  quarts  is  not  precisely  the  same  as  three- 
fourths  of  a  gallon,  though  they  are  equal  in  value.  In  the 
latter  case,  there  is  a  direct  and  necessary  relation  of  a  part  to 


CLASSES    OF    COMMON    FRACTIONS.  425 

a  unit ;  in  the  former  case,  no  such  relation  is  implied.  To  an- 
derstand  the  fraction,  three-fourths  of  a  gallon,  the  idea  of  the 
unit,  gallon,  must  be  in  the  mind;  in  three  quarts  no  such  con- 
dition is  necessary.  In  one  case  there  are  two  units  considered, 
the  gallon  and  the  fourth;  in  the  other  case  but  one  unit,  the 
quart, — not  considering  the  unit  of  the  pure  numbers,  three 
and  four  themselves.  Fourths  have  reference  to  the  integral 
unit,  and  always  imply  this  relation ;  quarts  have  no  reference 
to  gallons,  and  do  not  imply  gallons. 

Again,  the  fraction  three-fourths  may  be  used  entirely  dis- 
tinct from  any  denominate  unit,  and  in  this  case  it  must  be  an 
abstract,  not  a  denominate  number.  Two  is  one-fourth  of  eight; 
here  the  measure  of  this  relation,  one-fourth,  cannot  but  be  al> 
stract.  It  is  evident,  therefore,  that  a  fraction  is  not  a  denom- 
inate number.  There  are  abstract  and  denominate  fractions,  as 
there  are  abstract  and  denominate  integers. 


CHAPTER  III. 

TREATMENT   OF   COMMON  FRACTIONS. 

A  FRACTION  has  been  defined  as  a  number  of  the  equal 
parts  of  a  unit.  The  parts  into  which  the  unit  is  divided 
are  called  fractional  units.  A  fraction  may,  therefore,  also  be 
defined  as  a  number  of  fractional  units.  Fractions  are  divided, 
as  previously  stated,  into  Common  and  Decimal  Fractions. 

A  Common  Fraction  is  a  number  of  fractional  units  expressed 
with  a  numerator  and  a  denominator;  as  two-thirds,  written  §. 
The  denominator  of  a  fraction  denotes  the  number  of  equal 
parts  into  which  the  unit  is  divided.  The  numerator  of  a  frac- 
tion denotes  the  number  of  fractional  units  in  the  fraction.  A 
common  fraction  is  usually  expressed  by  writing  the  numerator 
above  the  denominator  with  a  line  between  them.  Care  should 
be  taken  not  to  define  the  denominator  as  the  "figure  below  the 
line,"  and  the  numerator  as  the  "figure  above  the  line ;"  and 
then  speak  of  multiplying  the  numerator  and  denominator. 
This  will  lead  one  to  suppose  that  figures  may  be  multiplied, 
rather  than  the  numbers  which  they  represent.  It  is  surpris- 
ing that  so  many  writers  upon  arithmetic  should  have  fallen 
into  this  error. 

Gases. — Fractions  admit  of  the  same  general  treatment  as 
integers;  we  therefore  have  the  same  fundamental  cases  in 
fractions  as  in  whole  numbers.  These  cases  are  all  embraced 
under  the  general  processes  of  Synthesis,  Analysis,  and  Com- 
parison. The  cases  of  synthesis  and  analysis  are  the  same  as 
in  whole  numbers.  To  perform  the  synthetic  and  analytic 
processes,  we  need  to  change  fractions  from  one  form  to  another ; 

(426) 


TREATMENT   OP   COMMON   FRACTIONS. 


427 


hence  Reduction  enters  largely  into  the  treatment  of  fractions. 
The  comparison  of  fractions  gives  rise  to  several  cases  called 
the  Relation  of  Fractions,  which  do  not  appear  in  whole  num- 
bers. The  various  cases  of  fractions  then  are ;  Reduction,  Ad- 
dition, Subtraction,  Multiplication,  Division,  Relation,  Com- 
position, Factoring,  Common  Divisor,  Common  Multiple,  In- 
volution, and  Evolution. 

A  complete  view  of  the  fundamental  processes  is  presented 
in  the  following  logical  outline.  Composition,  Factoring,  Invo- 
lution, and  Evolution,  presenting  no  points  different  from  those 
of  whole  numbers,  are  omitted  in  the  treatment.  The  other 
cases  arising  out  of  Comparison  apply  equally  to  integers  and 
fractions,  and  do  not  require  a  distinct  treatment. 

1.  Number  to  a  Fraction. 


Outline 

of  the 

Cases 

of 

Fractions. 


1.  Reduction. 


2.  Fraction  to  a  Number. 

3.  To  Higher  Terms. 


4.  To  Lower  Terms. 

5.  Compound  to  Simple. 

6.  Dissimilar  to  Similar. 

The  denominators  alike. 
The  denominators  unlike. 
The  denominators  alike. 
The  denominators  unlike. 
Fraction  by  a  Number. 
Number  by  a  Fraction. 
Fraction  by  a  Fraction. 
Fraction  by  a  Number. 
Number  by  a  Fraction. 
Fraction  by  a  Fraction. 

1.  Number  to  a  Number. 

2.  Fraction  to  a  Number. 

3.  Number  to  a  Fraction. 

4.  Fraction  to  a  Fraction. 
The  "Relation  of  Fractions"  is  a  new  division  of  the  subject 

of  fractions:  it  was  first  published  in  the  Normal  Written 
Arithmetic,  in  1863,  and  has  since  been  introduced  into  several 
other  works  on  written  arithmetic,  and  will  probably  be  gen 
erally  adopted. 


2.  Addition. 


3.  Subtraction. 


4.  Multiplication 


5.  Division. 


6.  Relation. 


{i 
{i 

■8 

8 


428  THE    PHILOSOPHY    OF    ARITHMETIC. 

Methods  of  Treatment. — There  are  two  methods  of  develop- 
ing the  subject  of  common  fractions,  which  may  be  distinguished 
as  the  Inductive  and  Deductive  methods.  These  two  methods 
are  entirely  distinct  in  principle  and  form;  and  the  distinction, 
being  new,  seems  worthy  of  special  attention. 

By  the  Inductive  Method,  we  solve  each  case  by  analysis, 
and  derive  the  rules,  or  methods  of  operation,  from  these  anal- 
yses, by  inference  or  induction.  The  method  is  called  induc- 
tive, because  it  proceeds  from  the  analysis  of  particular  problems 
to  a  general  method  which  applies  to  all  problems  of  a  given 
class.  The  solutions,  it  will  be  noticed,  are  independent  of  any 
previously  established  principles  of  fractions,  each  case  being 
treated  by  the  method  of  arithmetical  analysis  which  reasons 
to  and  from  the  Unit. 

To  illustrate  the  method  we  will  take  the  problem,  "In  f  how 
many  twentieths?"  We  analyze  this  as  follows:  One  equals 
f-g-,  and  \  equals  £  of  20  twentieths,  or  5  twentieths;  and  f 
equals  3  times  5  twentieths,  or  15  twentieths;  hence  f  equals 
!~J.  Now,  by  examining  this  solution,  we  see  that  we  multiply 
the  numerator  of  f  by  the  number  which  denotes  how  many 
times  four,  the  given  denominator,  equals  the  required  denom- 
inator, twenty,  which  is  the  same  as  multiplying  both  terms  of  f 
by  the  same  number,  five;  hence  we  derive  the  rule,  "to  reduce 
a  fraction  to  higher  terms,  multiply  both  terms  by  the  same 
number." 

For  another  illustration,  take  the  converse  of  this  problem, 
"In  -|-§  how  many  fourths?"  The  solution  is  as  follows:  One 
equals  f£,  and  \  equals  \  of  f-g-,  which  is  -^j-;  hence  £  of  the 
number  of  20ths  equals  the  number  of  4ths  ;  -J  of  15  is  3,  hence 
J!  equals  j.  This  is  the  analysis  of  the  problem;  we  then 
proceed  to  derive  a  rule  by  which  all  such  problems  may  be 
solved.  By  examining  this  analysis,  we  see  that  we  take  the 
same  part  of  the  numerator  for  the  numerator  of  the  required 
fraction  that  the  denominator  of  the  required  fraction  is  of  the 
denominator  of  the  given  fraction;  hence  we  derive  the  rule, 


TREATMENT  OF   COMMON   FRACTIONS.  429 

"to  reduce  a  fraction  to  lower  terms  divide  both  numerator  and 
denominator  by  the  same  number."  This  rule  is  thus  obtained 
by  analyzing  the  analysis ;  it  may  also  be  obtained  by  compar- 
ing the  two  fractions.  Thus,  comparing  j  and  |-f,  we  see  that 
3  equals  15  divided  by  5,  and  4  equals  20  divided  by  5 — that 
is,  both  divided  by  the  same  number — and  seeing  that  this- 
principle  holds  good  in  several  cases,  we  infer  the  rule. 

By  the  Deductive  Method  we  first  establish  a  few  general 
principles  by  demonstration,  and  then  derive  the  rules,  or 
methods  of  operation,  from  these  principles.  The  method  is 
called  deductive  because  it  proceeds  from  the  general  principle 
to  the  particular  problem.  To  illustrate  this  method,  let  us 
solve  the  same  problem,  "  Reduce  f  to  twentieths."  By  a  gen- 
eral proposition  which  we  assume  has  been  demonstrated,  we 
have  the  principle,  "Multiplying  both  terms  of  a  fraction  by  any 
number  does  not  change  its  value;"  hence  we  may  reduce  |  to 
twentieths  by  multiplying  both  terms  by  5,  which  will  give 
the  required  denominator,  and  we  have  f  equal  to  |-J. 

For  another  illustration,  we  will  solve  the  converse  problem, 
"Reduce  -||  to  fourths."  By  a  general  proposition,  which  we 
assume  has  been  demonstrated,  we  have  the  principle,  "  Divid- 
ing both  terms  of  a  fraction  by  the  same  number  does  not  chaoge 
its  value;"  hence  we  may  reduce  |~J  to  fourths  by  dividing 
both  numerator  and  denominator  by  any  number  which  will 
give  the  required  denominator.  This  number,  we  see,  is  5 ; 
hence,  dividing  both  numerator  and  denominator  by  5,  we 
have  |-|  equal  to  j. 

We  will  illustrate  the  difference  of  these  two  methods  still 
further  by  a  problem  in  compound  fractions.  Take  the  ques- 
tion, "What  is  §  of  |?"  The  analysis  is  as  follows:  ^  of  £  is 
one  of  the  three  equal  parts  into  which 4- may  be  divided;  if  each 
5th  is  divided  into  3  equal  parts,  -f  or  the  Unit  will  be  divided 
into  5  times  3,  or  15  equal  parts,  and  each  part  will  be^;  hence 
£  of  £  is  -j5-,  and  £  of  f  is  4  times  y1^,  or  y\,  and  §  of  |  is  2  times 
r\>  or  -jSg-.     Examining  this  analysis,  we  see  that  we  have  mul- 


430  THE   PHILOSOPHY    OF    ARITHMETIC. 

tiplied  the  two  denominators  together  and  the  two  numerators 
together,  from  which  we  derive  the  rule  for  the  reductiou  of 
compound  fractions.  By  the  deductive  method  we  would 
reason  as  follows :  By  a  principle  previously  demonstrated,  ^ 
of  f,  which  is  the  same  as  dividing  |  by  3,  is  T4g-;  and  f  of  f 
by  another  principle,  is  T8-.  It  will  be  noticed  that  the  deduc- 
tive method  is  much  shorter  than  the  inductive  method,  because 
while  the  former  explains  every  point  involved,  the  latter  makes 
use  of  principles  previously  demonstrated.  If  in  the  deductive 
solution,  we  should  stop  and  demonstrate  the  principles  we  are  to 
use,  it  would  make  the  solution  much  longer.  The  difference 
of  the  two  methods  can  also  be  clearly  illustrated  in  the  divi- 
sion and  relation  of  fractions.  In  my  higher  arithmetic  the  two 
methods  are  presented  in  each  case,  where  a  full  comparison 
may  be  made  of  them. 

The  distinction  between  these  two  methods  is  broad  and 
emphatic.  By  the  Inductive  Method  the  problem  is  solved 
without  any  reference  to  any  previously  established  principle  ; 
by  the  Deductive  Method,  the  solution  is  derived  from  a  gen- 
eral principle  supposed  to  have  been  previously  demonstrated. 
Both  of  these  methods  may  be  used  in  the  development  of  frac- 
tions, and  it  is  a  question  worthy  of  consideration  which  is  to 
be  preferred. 

The  Inductive  Method  is  believed  to  be  simpler  and  more 
easily  understood  by  young  pupils.  It  is  especially  adapted  to 
beginners,  since  it  proceeds  according  to  the  simple  steps  of 
analysis,  or  the  comparison  of  the  collection  with  the  unit.  It 
also  follows  the  law  of  the  development  of  the  young  mind — 
"from  the  particular  to  the  general."  It  is  especially  suited 
to  the  subject  of  Mental  Arithmetic,  on  account  of  its  simplicity 
and  the  mental  discipline  it  is  calculated  to  afford. 

The  Deductive  Method  is  more  difficult  in  thought  than  the 
Inductive  Method.  Young  pupils  always  find  a  difficulty  in 
founding  a  process  of  reasoning  upon  previously  established 
principles.    It  is  not  natural  for  the  youthful  mind  to  reason  from 


TREATMENT   OF    COMMON   FRACTIONS.  4:31 

generals  to  particulars.  Besides,  the  demonstrations  of  these 
general  principles  are  not  readily  understood  by  young  pupils. 
With  much  experience  as  a  teacher,  I  state  that  it  is  n.  rare 
thing  to  find  a  pupil  who  can  give  a  good  logical  demonstration 
of  these  principles,  aad  text-books  and  teachers  often  do  no 
better.  The  so-called  demonstrations  in  many  of  our  text-books 
are  mere  explanations  or  illustrations,  and  not  logical  proofs  of 
the  propositions.  To  say  that  "  multiplying  the  denominator 
of  a  fraction  increases  the  number  of  parts  of  the  fraction,  and 
diminishes  their  size  in  the  same  proportion,"  is  a  loose  sort  of 
statement  that  comes  very  far  short  of  scientific  demonstration. 
We  will  consider  these  principles  and  their  demonstration. 

Fundamental  Principles.— In  the  Deductive  Method,  we 
have  stated,  we  first  establish  several  general  principles,  and 
then  derive  the  rules  or  methods  of  operation  from  them. 
These  principles  relate  to  the  multiplication  of  the  numerator 
and  denominator  of  a  fraction.  They  may  be  demonstrated  in 
two  distinct  ways.  One  of  these  is  founded  upon  the  princi- 
ples of  division  ;  the  other  upon  the  nature  of  the  fraction  and 
the  functions  of  the  numerator  and  denominator.  All  the 
various  methods  in  our  text-books  on  arithmetic  may  be  em- 
braced under  these  two  general  methods. 

The  Method  of  Division  is  employed  by  a  large  majority  of 
our  writers  on  arithmetic.  This  method  consists  in  regarding 
the  fraction  as  an  expression  of  an  unexecuted  division,  the 
numerator  representing  the  dividend,  and  the  denominator  the 
divisor,  and  the  value  of  the  fraction  being  the  quotient.  Then, 
by  principles  of  division  presumed  to  have  been  previously 
established,  since  dividing  the  dividend  divides  the  quotient, 
dividing  the  numerator  divides  the  fraction;  and  since  multi 
plying  the  divisor  divides  the  quotient,  multiplying  the  denom 
inator  divides  the  fraction,  etc. 

The  Fractional  Method  of  demonstrating  these  fundamental 
principles  is  based  upon  the  nature  of  the  fraction  itself.  It 
regards  the  fraction  as  a  number  of  equal  parts  of  a  unit,  am* 


432  THE    PHILOSOPHY    OP   ARITHMETIC. 

determines  the  result  of  these  operations  by  comparing  the 
fractional  unit  with  the  Unit.  Thus,  if  we  multiply  the  de- 
nominator of  a  fraction  by  any  number,  as  three,  the  Unit  will 
be  divided  into  three  times  as  many  equal  parts,  hence  each 
part  will  be  one-third  as  large  as  before;  and  the  same  number 
of  parts  being  taken,  the  value  of  the  fraction  will  be  one-third 
as  large  as  before.  In  a  similar  manner  all  the  principles  may 
be  demonstrated. 

The  Fractional  Method  is  undoubtedly  the  correct  one.  The 
Method  of  Division  is  liable  to  several  objections,  and  should 
be  discarded  in  teaching  and  in  writing  text-books,  as  appears 
from  several  considerations. 

First,  it  is  illogical  to  leave  the  conception  of  a  fraction  and 
pass  to  that  of  division,  to  establish  a  principle  of  a  fraction. 
A  fraction  and  an  expression  of  division  are  two  distinct 
things,  and  should  not  be  confounded.  The  fraction  J  is 
three-fourths,  and  does  not  mean  3  divided  by  4.  It  is  true 
that  the  expression  f  does  also  mean  3  divided  by  4 ;  but  when 
we  regard  it  as  a  fraction  we  have  and  should  have  no  idea  of 
the  division  of  three  by  four.  It  is,  therefore,  illogical,  I  say,  to 
convert  a  fraction  into  a  division  of  one  number  by  another  to 
attain  to  a  principle  of  the  fraction. 

Secondly,  it  is  not  only  illogical  to  treat  the  subject  in  this 
manner,  but  it  does  not  give  the  learner  the  true  idea  of  it. 
lie  may  see  that  multiplying  the  denominator  does  divide  the 
value  of  the  fraction,  but  he  will  not  see  down  into  the  core  of 
the  matter,  why  it  does  so.  The  method,  to  say  the  least, 
gives  but  a  superficial  view  of  the  subject,  and  is  therefore 
objectionable.  If  the  fraction  will  admit  of  a  simple  treatment 
as  a  fraction,  it  is  absurd  to  transform  it  into  something  else 
to  prove  its  principles. 

It  may  be  said  in  favor  of  the  method  of  division,  that  it  is 
simpler  and  more  easily  understood  by  ,.«  I-earner;  but  this 
both  theory  and  experience  in  instruction  will  disprove.  I 
believe  that  the  pupil  can  quite  as  readily  see  tbut  dividing 


TREATMENT    OF   COMMON   FRACTIONS.  433 

the  numerator  of  a  fraction  divides  the  value  of  the  fraction,  as 
he  can  see  that  dividing  the  dividend  divides  the  quotient;  and 
the  same  holds  for  the  other  principles.  This  method  may 
sometimes  seem  a  little  easier  to  the  learner,  because  it 
depends  upon  an  assumed  principle;  but  require  the  pupil  to 
prove  that  principle,  and  he  will  find  it  quite  as  difficult  as  to 
prove  the  fractional  principle  itself.  For  the  method  of  demon- 
strating these  theorems  which  the  author  prefers  in  arithmetic, 
the  reader  is  referred  to  his  arithmetical  works. 


CHAPTER   IT. 

CONTINUED   FRACTIONS. 

EVERY  new  idea,  when  once  fixed,  becomes  a  starting  point 
from  which  we  pass  to  other  new  ideas.  The  mind  never 
rests  satisfied  with  the  old;  it  is  always  reaching  out  beyond 
the  known  into  the  unknown.  "  Still  sighs  the  world  for  some- 
thing new,"  is  as  true  in  science  as  in  society.  Given  a  new 
conception,  and  the  tendency  is  to  push  it  forward  until  it  leads 
us  to  other  ideas  and  truths  not  anticipated  in  the  original  con- 
ception. Thus,  from  the  original  idea  of  a  simple  fraction 
originated  the  compound  and  complex  fractions;  and  thus  also 
by  extending  the  original  conception,  arose  the  Continued  Frac- 
tion. 

Definition A  Continued  Fraction  is  a  fraction  whose  nu- 
merator is  1,  and  denominator  an  integer  plus  a  fraction  whose 
numerator  is  also  1  and  denominator  a  similar  fraction,  and  so 
on.     Thus,  Jh=Wr     ' 

Several  recent  authors,  for  convenience,  write  a  continued 
fraction  with  the  sign  of  addition  between  the  denominators; 

thus.  —     —     —    — 
'   2  +  3  +  4   f  5 

Origin. — Continued  Fractions  were  first  suggested  to  the 
world  in  a  work  by  Cataldi,  published  in  1613,  at  Bologna. 
Cataldi  reduces  the  square  roots  of  even  numbers  to  continued 
fractions,  and  then  uses  these  fractions  in  approximation,  though 
without  the  modern  rule  by  which  each  approximation  is  educed 
from  the  preceding  two.  Daniel  Sch wenter,  according  to  Fink, was 
the  first  to  make  any  material  contribution  (1G25)  towards  deter- 
mining the  convergents  of  continued  fractions.  Continued  fractions 
were  also  proposed  about  the  year  1670,  by  Lord  Brouncker, 
President  of  the  Royal  Society.  It  is  known  that  in  order  to  ex- 
press the  ratio  of  the  circumscribed  square  to  the  circumference 

(434) 


CONTINUED     FRACTIONS.  435 

of  the  circle,  he  derived  the  following  con-     1+1 — 

tinued  fraction  given  in   the  margin  ;  but  *+^5_  ^ 

by  what  means  he  was  led  to  it,  has  not 

been  ascertained.     He  was  the  first  to  investigate  and  make 

any  use  of  their  properties.     Dr.  Wallis  subsequently  added  to 

and  improved  the  subject,  giving  a  general  method  of  reducing 

all  kinds  of  continued  fractions  to  common  fractions. 

The  complete  development  of  these  fractions,  with  their  ap- 
plication to  the  solution  of  numerical  equations  and  problems 
in  indeterminate  analysis,  is  due  to  the  Continental  mathemati- 
cians. Huygens  is  said  to  have  explained  the  manner  of  form- 
ing the  fractions  by  continual  divisions,  and  to  have  demon- 
strated the  principal  properties  of  the  converging  fractions 
which  result  from  them.  John  Bernoulli  made  a  happy  and 
useful  application  of  the  continued  fraction  to  a  new  species  of 
calculation  which  he  devised  for  facilitating  the  construction  of 
tables  of  proportional  parts.  The  most  complete  development 
of  continued  fractions  was  given  by  Euler,  who  introduced  the 
term  f radio  conlinua. 

Treatment. — The  subject  of  continued  fractions  is  most  con- 
veniently treated  by  the  algebraic  method,  and  may  be  found 
quite  fully  presented  in  some  of  the  works  on  higher  algebra. 
In  this  place  we  shall  briefly  consider:  1.  Reducing  common 
fractions  to  continued  fractions;  2.  Reducing  continued  frac- 
tions to  common  fractions;  3.  Their  application ;  4.  Their  prin- 
ciples. 

We  shall  first  show  how  a  common  fraction  may  be  reduced 
to  a  continued  fraction.   Take  the  common  fraction  fgj.  Dividing 
both  numerator  and  denominator  by  G8, 
we  have  the  first  expression  in  the  mar-     2^.21.         2qri_g 
gin;  dividing  the  numerator  and  denom-  +  ?i 

inator  of  the  second  fraction  by  21,  we  £qpi 

have  the  second  expression  in  the  margin;  ^FJ 

dividing  again  by  5,  we  have  the  third 

expression  in  the   margin;    which   finishes  the   division,  as 


486  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  numerator  of  the  last  fraction  is  unity.  The  terms  £,  J.  £, 
etc.,  are  called  the  first,  second,  third,  etc.,  partial  fractious. 
-  The  same  result  may  be  obtained  by  dividing  as  iu  finding 
the  greatest  common  divisor,  and  taking  the  several  quotients 
for  the  successive  denominators.  Taking  T6^r,  and  dividing  as 
if  to  find  the  greatest  common  divisor  of  its  terms, 
we  see  that  the  resulting  quotients  are  the  same  as  68J15T 
the  denominators  of  the  partial  fractions.    Hence  we     zz    ^ 


derive    the   following  rule   for   reducing  common       ^ 
fractions  to  continued  fractions :     Find  the  greatest 


21 

20 

1 


common  divisor  of  the  terms  of  the  given  fraction  ; 

the  reciprocals  of  the  successive  quotients  will  be  the  partial 

fractions  which  constitute  the  continued  fraction  required. 

Let  us  now  see  how  a  continued  fraction  may  be  reduced  to 
a  common  fraction.  This  reduction  may  be  effected  in  two 
ways;  by  beginning  at  the  last  fraction  and  working  up, 
or  by  beginning  at  the  first  fraction  and  working  down. 

If  we  take    the   continued   fraction    given    in   the   margin 

and    reduce     the    complex    fraction    formed 

by   the   last   two   terms    to    a    simple    frac-     3~i~r 

tion,  we  shall   have  -fa.     Taking  this  result  +2qrj_. 

and  the  preceding  partial   fraction  together,  • 

15  21 

we  have  —     — -,  which  reduced  equals  — .     Joining  this  to  the 
2  -f-  21  47 

preceding  term,  we  have  —    — — ,  which  equals  — .     Finally, 

—     — -— =a-— — — - ,  the  value  of  the  fraction. 
o  -p  bo       251 

By  beginning  at  the  first  fraction,  approximate  values  of  the 

continued  fraction  may  be  obtained  by  respectively  reducing 

two,  three,  or  more  of  the  partial  fractions  to  simple  fractions. 

Thus,  in  the  fraction  given  above,  the  first  approximate  value 

•ii-  ,!!  1,1..  .3  •      Ill 

is  £ ;  the  second  — -     — -,  or  — ;  the  third  is  —     — -     — -,  or 

o-f-1  4  o-f-l~r2 

~;  the  fourth  i|;  the  fifth  ^-. 


CONTINUED   FRACTIONS.  437 

By  exhibiting  this  process  in  an  analytic  form,  a  law  may  be 
discovered  which  presents  a  simpler  and  easier  method  of  find- 
ing approximate  values  than  either  of  the  others. 
Let  us  take  the  fraction  in  the  margin  and  find     2  1  1 
its  successive  approximate  values,  and  notice  the  Mh^ 

la^   jf  the  derivation  of  one  approximation  from 
the  previous  ones.     The  work  may  be  written  as  follows: 
-]  =\,  1st  approx.  val. 

hrr= —  =f,2d 

*+i     3x2+1 

1 


+i+|=2+5 =         3x5+1 


3x5+1 


3x5+1        (3x2+l)X5+2  -7x5+2  -ir,-d 

1 

:2+5+i _ (3X5+l)X4+3 _ 

3x(5+l)+l         {  (3X2+l)X5+2  }X  4+3x2+1 
10x4+3 
37x4+7      153' 

We  take  -J,  the  first  term  of  the  continued  fraction,  for  the  first 
approximate  value.  Reducing  the  complex  fraction  formed  by 
the  first  two  terms  of  the  continued  fraction,  we  have  f  for  the 
second  approximate  value.  Continuing  the  reduction,  we 
obtain  44  and  ■£?-?■  for  the  remaining  values.  Examining  the 
last  two  reductions,  we  find  that  the  third  approximate  value 
is  obtained  by  multiplying  the  terms  of  the  second  approximate 
fraction  by  the  denominator  of  the  third  partial  fraction,  and 
adding  to  these  products  the  corresponding  terms  of  the  first 
approximate  fraction.  We  see  also  that  the  fourth  approximate 
value  is  equal  to  the  product  of  the  terms  of  the  third  approxi- 
mate value  by  the  denominator  of  the  fourth  partial  fraction, 
plus  the  corresponding  terms  of  the  second  approximation. 
Hence  we  derive  the  following  rule : 

For  the  first  approximate  value  take  the  first  partial 
fraction;  for  the  second  value,  reduce  the  complex  fraction 
formed  by  the  first  two  terms  of  the  continued  fraction  ;  for 
each  succeeding  approximate  value,  multiply  both  terms  of 


+*+? 


438  THE   PHILOSOPHY   OF    ARITHMETIC. 

the  approximation  last  obtained  by  the  next  denominator  of 
the  continued  fraction,  and  add  to  the  products  the  corre- 
sponding terms  of  the  preceding  approximation. 

TVe  will  now  show  the  application  of  continued  fractions  by 
the  solution  of  several  practical  questions. 

1.  Let  it  be  required  to  express  approximately,  in  the  fraction 
of  a  day,  the  difference  between  a  solar  year  and  365  days. 

By  the  old  reckoning,  the  excess  of  the  solar  year  over  365 
days  was  5  hours,  48  minutes,  48  seconds.  Reducing,  we  find  this 
excess  equals  20,928  seconds,  and  24  hours  equals  86,400  seconds. 
Therefore,  the  true  value  of  the  fraction =|JJJ|.=J£j{..  Now, 
converting  \^  into  a  continued  fraction,  we  have  the  expres- 
sion given  in  the  margin,  from  which, 
by  the  last  rule,  we  obtain  the  approx-  ttt=s 
imate  values  £,  ¥7¥,  ft,  T\\,  -&\,  J£$. 

The  fraction  £  agrees  with  the 
correction  introduced  into  the  calendar  by  Julius  Caesar,  by 
means  of  bissextile  or  leap  year.  The  fraction  ft  is  the  cor- 
rection used  by  the  Persian  astronomers,  who  add  8  days  in 
every  33  years,  by  having  7  regular  leap-years,  and  then  de- 
ferring the  eighth  for  5  years. 

2.  Required  the  approximate  ratio  of  the  English  foot  to 
the  French  metre  containing  39.371  inches. 

The  true  ratio  is  -J-f-f  S-0#  Reducing  to  a  continued  fraction, 
we  find  some  of  the  first  approximate  values  to  be  ^,  ft,  ^ 
ft,  ff,  t3qV  Hence  the  foot  is  to  the  metre  as  3  to  10,  nearly; 
a  more  correct  ratio  is  32  to  105. 

3.  To  find  some  of  the  approximate  values  of  the  ratio  of 
*,he  circumference  of  a  circle  to  the  diameter. 

Taking  the  value  of  the  circumference  of  the  circle  whose 
diameter  is  1,  to  10  places  of  decimals,  the  ratio  of  the  diameter 
to  the  circumference  will  be  expressed  by  the  common  fraction 
1?tIt!HHHHH&  deducing  to  a  continued  fraction,  some  of  the 
first  approximate  values  are,  -J,  ft,  iff,  |4|.  Inverting  these 
tractions,  we  have  the  ratio  of  the  circumference  to  the  diame- 


CONTINUED   FRACTIONS.  439 

ter,  which  is  the  ratio  commonly  used.  The  second  gives  ^ 
which  is  the  ratio  said  to  have  been  found  by  Archimedes ;  and 
the  fourth  gives  f-f|,  which  is  the  same  as  that  determined  by 
Metius,  which  is  more  exact  than  3.141592,  from  which  it  is  de- 
rived. 

Continued  fractions  have  been  employed  for  obtaining  eloganl 
approximations  to  the  roots  of  surds.  Thus,  let  it  be  required 
to  find  the  square  root  of  J,  or  the  ratio  of  the  side  of  a  square 
to  its  diagonal. 

The  square  root  of  i,  or  </\,  equals  — -.     Dividing  both 

terms  by  the  numerator  we  have  — —  = .      Multi- 

v/  2        l-}-^/2 — 1 

~~ 1 
v/2— 1 
plying  both  terms  of  the  fraction  — by  v/2+1,  it  will  be 

come    yo_L     =o-l    /o — r     Substituting,  we  have 
v/  JtI       Z  ~r  v  " —  1 

1 
1  1 

v/2  ""1+1 


2-fV2-l' 


v/2— 1 

Again,  the  fraction  — becomes,  as  before,  equal  to 


2+v/2-l 


1 

and  by  thus  continuing  the  process,  we  find  — —  to  equal  the 

v/2 

following  continued  fraction : 


*R 


+*+\ 


2+etc. 
Some  of  the  first  approximate  values  of  this  fraction  are  ■}•,  §, 

f  if  tt.  if  HI  etc. 

Continued  fractions  are  also  applied  to  the  solution  of  inde- 
terminate problems,  as  may  be  seen  in  Barlow's  Theory  of 
Numbers,  or  Legendre's  Theorie  des  Nombre*. 


440  THE   PHILOSOPHY    OF    ARITHMETIC. 

There  are  several  beautiful  principles  belonging  to  the  ap- 
proximate values  of  continued  fractions,  a  few  of  which  we 
present  in  this  place.  The  values  just  obtained  for  the  ratio 
of  the  side  of  a  square  to  its  diagonal  are  used  as  illustrations. 

1.  The  approximate  fractions  are  alternately  too  small  and 
too  large.  Thus,  f,  ff,  $#,  are  too  small,  while  -*-,  f,  f-J,  and 
Jf-f  are  too  large. 

2.  Any  one  of  these  fractions  differs  from  the  true  value 
of  the  continued  fraction  by  a  quantity  which  is  less  than  the 
reciprocal  of  the  square  of  its  denominator.  Thus,  j-f ,  which 
is  the  ratio  much  used  by  carpenters  in  cutting  braces,  differs 
from  the  true  ratio  by  a  quantity  less  than  (tt)2=2  8¥- 

3.  Any  two  consecutive  approximate  fractions,  when  re 
duced  to  a  common  denominator,  will  differ  by  a  unit  in  their 
numerators.  Thus  f  and  -J-J-,  when  reduced  to  a  common  de- 
nominator, become  -j^  and  -j^. 

4.  All  approximate  fractions  are  in  their  lowest  terms.  If 
they  were  not,  the  difference  of  the  numerators  of  two  consec- 
utive approximate  fractions,  when  reduced  to  a  common  denom- 
inator, would  differ  by  more  than  unity.  For  each  numerator 
is  multiplied  by  the  denominator  of  the  other  fraction,  hence 
one  derived  numerator  contains  the  original  numerator,  and 
the  other  the  original  denominator  of  either  fraction.  If  then 
there  were  a  common  factor,  it  must  be  a  factor  of  the  difference 
of  the  numerators;  and  this  difference  would  be  greater  than 
unity,  which  is  contrary  to  the  previous  principle. 

The  successive  approximate  values  are  called  the  convergents 
of  the  fraction.  The  numerator  or  denominator  of  the  convergent 
is  called,  by  Sylvester,  a  cumulant.  A  non-terminating  contin- 
ued fraction  whose  quotients  recur,  is  called  a  periodical  or 
recurring  continued  fraction.  Its  value  can  be  shown  to  be 
equal  to  one  of  the  roots  of  a  quadratic  equation.  It  can  also 
be  shown  that  every  quadratic  surd  gives  rise  to  an  equivalent 
periodic  continued  fraction. 


SECTION   II. 

DECIMAL  FRACTIONS. 


iy* 


I.  Origin  of  Decimals.. 


II.  Treatment  op  Decimals. 


III.  Nature  op  Circulates. 


IY.  Treatment  op  Circulates. 


V.  Principles  of  Circulates. 


VI.  Complementary  Repetends. 


VII.  A  New  Circulate  Form. 


CHAPTER  L 

ORIGIN  OF   DECIMALS. 


THE  invention  of  the  Decimal  Fraction,  like  the  invention  of 
the  Arabic  scale,  was  one  of  the  happy  strokes  of  genius. 
The  common  fraction  was  expressed  by  a  notation  quite  dis- 
tinct from  that  of  integers,  and  required  not  only  a  different 
treatment,  but  one  much  more  complicated  and  difficult.  The 
expression  of  the  decimal  divisions  of  the  unit  in  the  same  scale 
with  integers,  and  the  possibility  of  reducing  common  fractions 
to  the  decimal  form,  wrought  quite  a  revolution  in  the  science 
of  arithmetic,  and  has  greatly  simplified  it.  This  new  method 
of  expressing  fractions  gave  rise  to  a  much  simpler  method  of 
treating  them ;  and  has  elevated  the  decimal  fraction  into  dis- 
tinction, and  gained  for  it  an  independent  consideration. 

Origin. The  Decimal  Fraction  had  its  origin  within  the  last 

three  centuries.  Theoretically  it  may  have  originated  in  either 
of  two  ways.  There  may  have  been  a  transition  from  the  com- 
mon fraction  to  the  decimal,  by  noticing  that  a  number  of  tenths, 
hundredths,  etc.,  might  be  expressed  by  the  decimal  scale. 
This  is  the  manner  in  which  the  subject  is  usually  presented  in 
the  text-books  of  the  present  day.  Thus,  after  the  pupil  is  made 
familiar  with  the  fractions  ^  rro>  etc->  'lt  is  stated  that  A  ma7 
be  expressed  thus,  .1 ;  yfo  thus,  .01,  etc.  The  decimal  fraction 
could  also  have  arisen  directly  from  the  decimal  scale.  Thus, 
since  the  law  of  the  scale  is,  that  terms  diminish  in  value  from 
left  to  right  in  a  ten-fold  ratio,  the  idea  of  carrying  the  scale  on 
to  the  right  of  the  unit  would  naturally  present  itself,  and  such 
a  continuation  would  give  rise  to  the  decimal.     As  the  unit 

(443) 


444  THE    PHILOSOPHY    OF    ARITHMETIC. 

was  one-tenth  of  the  tens,  the  first  place  to  the  right  of  the 
unit  would  be  one-tenth  of  the  unit;  the  second  place,  one-tenth 
of  one-tenth,  or  one-hundredth,  etc.  These  two  methods  of 
conceiving  the  origin  of  decimals  are  entirely  distinct;  indeed, 
they  are  the  converse  of  each  other.  In  one  case  we  pass  from 
the  common  fraction  to  its  expression  in  the  decimal  scale;  in 
the  other  we  pass  from  the  expression  in  the  decimal  scale 
to  the  fraction.  This  distinction,  it  may  be  remarked,  has  a 
practical  bearing  upon  the  method  of  teaching  the  subject.  In 
which  way  it  did  actually  originate  is  not  definitely  known, 
though  De  Morgan  holds  that  the  table  of  compound  interest 
suggested  decimal  fractions  to  Stevinus. 

History. — The  introduction  of  decimal  fractions  was  formerly 
ascribed  to  Regiomontanus,  but  subsequent  investigations  have 
shown  this  to  be  incorrect.  The  mistake  seems  to  have  arisen 
from  the  confused  manner  in  which  Wallis  stated  that  Regio- 
montanus introduced  the  decimal  radius  into  trigonometry  in 
place  of  the  sexagesimal.  Decimal  fractions  were  introduced  so 
gradually  that  it  is  difficult,  if  not  impossible,  to  assign  their 
origin  to  any  one  person.  The  earliest  indications  of  the  deci- 
mal idea  are  found  in  a  work  published  in  1525  by  a  French 
mathematician  named  Orontius  Fineus.  In  extracting  the 
square  root  of  10,  he  extracts  the  approximate  root  of  10000000 
and  obtains  31G2.  He  then  separates  162,  which  with  him  is 
not  a  fraction,  but  only  a  means  of  procuring  fractions,  and 
converts  it,  after  the  scientific  custom  of  the  times,  into  sexa- 
gesimal fractions  (having  as  base  60),  so  that  the  square  root 
of  10  would  be  expressed  3  9'  43"  12"',  or  S+^+nVh+Trlhr* 
He  concludes  that  chapter  of  his  book  by  stating  that  in  this 
162, 1  is  a  tenth,  6  is  six  hundredths,  etc.,  so  that  it  would  seem 
that  he  had  quite  a  clear  notion  of  decimals. 

Tartaglia,  in  1556,  gives  a  full  account  of  the  metnod  of 
Orontius,  but  prefers  the  common  fractional  form  Sj1^6^.  In 
Recorde's  Whetstone  of  Witte,  1557,  the  same  rule  is  copied; 
but  after  obtaining  three  decimal  places  of  the  square  root,  the 


ORIGIN   OF    DECIMALS.  445 

remainder  is  written  as  a  common  fraction.  Peter  Ramus,  in 
an  arithmetic  published  in  Paris  in  1584  or  1592,  also  quotes 
the  rule  of  Orontius. 

In  1585,  Stevinus  wrote  a  special  treatise  in  French,  called 
"  The  Disme,  by  the  which  we  can  operate  with  whole  numbers 
without  fractions."  It  was  first  published  in  Dutch  about  the 
year  1590,  and  describes  in  very  express  and  simple  terms  the 
advantages  to  be  derived  from  this  new  arithmetic.  Decimals 
are  called  nombres  de  disme:  those  in  the  first  place  whose  sign 
is  (1)  are  called  primes,  those  in  the  second  place  whose  sign  is 
(2)  are  called  seconds,  and  so  on  ;  whilst  all  integers  are  char- 
acterized by  the  sign  (0),  which  is  put  over  the  last  digit.  The 
following  are  some  of  his  arithmetical  operations  by  means  of 
decimals,  representing  multiplication  and  division. 

(0)0)  (2)  (0)  (1)(2)  (3)  (4)  (5)  (1)  (2) 

3257  344352  (9  6 

8  9  4  6 


19  5  4  2  1 

13028  186 

29313  5114 

26056  7637      (omi)(2)(3) 

29  13712  2  3  4  4  3  5  2(3  5  8  7 

9  6  6  6  6 
9  9  9 

It  will  be  seen  that  he  employs  the  "  scratch  method  "  of 
division.  The  following  is  an  example  of  indefinite  division 
found  in  his  work : 

(0)  (1)  (2)  (3) 

|=1  3  3  3 
In  this  treatise  Stevinus  proposed  to  supersede  fractions  by 
dismes,  or  decimals.  He  enumerates  the  advantages  which 
would  result  from  the  decimal  subdivision  of  the  units  of  length, 
area,  capacity,  value,  and  lastly  of  a  degree  of  the  quadrant, 
in  the  uniformity  of  notation,  and  the  increased  facility  of  per- 
forming all  arithmetical  operations  in  which  fractions  of  such 
units  were  involved.  It  is  remarkable,  however,  that  though 
while  he  confines  himself  to  the  matter  of  his  computation  he 


446  TUE    PHILOSOPHY    OF    ARITHMETIC. 

admits  his  dismes,  when  he  passes  to  their  form  he  converts 
them  into  integers.  Still,  he  must  be  regarded  as  the  real  in- 
ventor and  introducer  of  the  system  of  decimals.  De  Morgan 
says  "The  Disme  is  the  first  announcement  of  the  use  of  deci- 
mal fractions ;"  and  Dr.  Peacock  also  remarks  that  "the  first 
notice  of  decimals,  properly  so  called,  is  to  be  found  in  La 
Disme." 

This  work  of  Stevinus  was  translated  into  English  in  1608, 
by  Richard  Norton,  under  the  title,  "Disme,  the  arte  of  tenths, 
or  decimal  Arithmetike,  teaching  how  to  perform  all  computa- 
tions whatsoever  by  whole  numbers  without  fractions,  by  the 
four  principles  of  common  Arithmetike  :  namely,  addition,  sub- 
traction, multiplication,  and  division,  invented  by  the  excellent 
mathematician,  Simon  Stevin."  In  this  work  the  notation  is 
changed  to 

(1)    (2)    (3)    (4) 

3,  7,  5,  9. 
The  introduction  of  decimals  into  works  on  arithmetic  was 
slow,  even  after  their  use  had  been  shown  by  Stevinus.  One 
of  the  earliest  English  works  in  which  decimal  fractions  are 
really  used,  is  that  of  Richard  Witt,  1613,  containing  tables 
of  half-yearly  and  compound  interest.  These  tables  are  con- 
structed for  ten  million  pounds ;  seven  figures  are  cut  off,  and 
the  reduction  to  shillings  and  pence,  with  a  temporary  decimal 
separation,  is  introduced  when  wanted.  Thus,  when  the  quar- 
terly table  of  amounts  of  interest  at  ten  per  cent,  is  used  for 
three  years,  the  principal  being  100Z.,  in  the  table  stands  1372- 
66429,  which  multiplied  by  100  and  seven  places  cut  off,  gives 
tne  first  line  of  the  following  citation : 
"  The  Worke 

(1         1372     66429 
Faoit  \  sh  13     2858 

(d  3     4296." 

Giving  1372Z.  13s.  3d.  for  the  answer.  The  tables  are  expressly 
stated  to  consist  of  numerators,  with  100...  for  a  denominator. 
Napier's  work,  published  in  1617,  contains  a  treatise  on  deci- 


ORIGIN     OF    DECIMALS.  447 

mals,  though  he  does  not  use  the  decimal  point,  except  in  one 
or  two  instances,  but  rather  indicates  the  place  of  the  decimal 
figures  by  primes,  seconds,  etc.,  according  to  the  method  of 
Stevinus.  The  author  expressly  attributes  the  origin  of  dec- 
imals to  Stevinus. 

In  1619  we  find  the  contents  of  Norton's  treatise  embodied 
in  an  English  work  entitled,  "The  Art  of  Tens,  or  Decimal] 
Arithmetike,  wherein  the  art  of  Arithmetike  is  taught  in  a 
more  exact  and  perfect  method,  avoyding  the  intricacies  of 
fractions.  Exercised  by  Henry  Lyte,  Gentleman,  and  by  him 
set  forth  for  his  countries  good.  London,  1619."  It  is 
dedicated  to  Charles,  Prince  of  Wales,  and  he  tells  us  that  he 
has  been  requested  for  ten  years  to  publish  his  exercises  in 
decimall  Arithmetike.  After  enlarging  upon  the  advantages 
which  attend  the  knowledge  of  this  arithmetic  to  landlords 
and  tenants,  merchants  and  tradesmen,  surveyors,  gaugers, 
farmers,  etc.,  and  all  men's  affairs,  whether  by  sea  or  land,  he 
adds,  "if  God  spare  my  life,  I  will  spend  some  time  in  most 
cities  of  this  land  for  my  countries  good  to  teach  this  art." 
This  author  was  one  of  the  earliest  users  of  decimal  fractions 

In  the  year  1619  there  appeared,  at  Frankfort,  a  work  on 
decimal  arithmetic  by  Johann  Hartman  Beyern,  in  which  the 
author  states  that  he  first  thought  upon  the  subject  in  the  year 
1597,  but  that  he  was  prevented  from  pursuing  it  for  many 
years  by  the  little  leisure  afforded  him  from  his  professional 
pursuits.  He  makes  no  mention  of  Stevinus,  but  assumes 
throughout  the  invention  as  his  own.  The  decimal  places 
are  indicated  by  the  superscription  of  the  Roman  numerals, 
though  the  exponent  corresponding  to  every  digit  in  the 
decimal  places  is  not  always  put  down.  Thus,  34.1426  is 
written  34°.1I411  2m6IV,  or  34°.14n26IV,  or  34°.1426IV. 
The  author  must  have  been  acquainted  with  the  Babdologia 
of  Napier,  as  one  chapter  of  his  work  is  devoted  to  the 
explanation  of  the  construction  and  use  of  these  rods,  which 
enjoyed  a  most  extraordinary  popularity  at  that  period;  and 


448  TIIE    PHILOSOPHY    OF    ARITHMETIC. 

he  could  not,  consequently,  have  been  ignorant  of  Napier's 
notation  or  of  the  work  of  Stevinus ;  and  we  may  therefore 
doubt  the  truth  of  his  pretensions  to  being  the  originator  of 
the  system  of  decimals. 

Albert  Girard  published  an  edition  of  the  works  of  Stevinus 
in   1625,  and  in  the  solution  of  the  equation  x3  —  3x—  1  by  a 
table  of  sines,  of  which  method  he  was  the  author, 
we  find   the  three  roots   as   in   the   margin.     On         1,532 1 

another  occasion,  he  denotes  the  separation  of  the     .     ^    f 

integers   and   decimals   by   a   vertical   line.       He 
does  not  always  adhere  to  this  simple  notation,  as  we  after- 
wards find  the  square  root  of  4  J  expressed  by  20816(4);  and 
on  another  occasion  we  find    similar  vestiges  of  the  original 
notation  of  Stevinus. 

Oughtred  is  said  to  have  contributed  much  to  the  propaga- 
tion and  general  introduction  of  decimal  arithmetic.  In  the 
first  chapter  of  his  Glavis,  published  in  1631,  we  find  an 
explanation  of  decimal  notation.  The  integers  he  separates 
from  the  decimal,  or  parts,  by  a  mark,  L,  which  he  calls  the 
separatrix,  as  in  the  examples,  0156,  48|5,  for  .56  and  48.5; 
and  in  giving  examples  of  the  common  operations  of  arithmetic 
he  unites  them  under  common  rules.  His  view  of  the  theory 
of  decimals  was  generally  adopted,  and  in  some  cases  hi? 
notation  also,  by  English  writers  on  arithmetic  for  more  than 
thirty  years  after  this  period. 

In  "  Webster's  tables  for  simple  interest,"  etc.,  1634,  decimals 
seem  to  be  treated  as  a  thing  generally  known,  though  no 
decimal  point  is  used.  During  the  same  year,  1634,  Peter 
Herigone,  of  Paris,  published  a  work  in  which  he  introduces 
the  decimal  fraction  of  Stevinus,  having  a  chapter  "  des  nombres 
de  ia  dixme."  The  mark  of  the  decimal  is  made  by  marking 
the  place  where  the  last  figure  comes.  Thus  when  137  livres 
16  sous  is  to  be  taken  23  years  7  months,  the  product  of  1378' 
and  23583"'  is  found  to  be  32497374"",  or  3249  liv.  14  sous, 
8  deniers.     In  1633,  John  Johnson  (Survaighor)  published  a 


ORIGIN   OF   DECIMALS.  449 

work,  the  second  part  of  which  is  called  "Decimall  Arithmatick 
wherby  all  fractionall  operations  are  wrought,  in  whole  num- 
bers," etc.  In  his  decimal  fractions  Johnson  has  the  rudest 
form  of  notation  ;  for  he  generally  writes  the  places  of  decimals 

I    1.2.3.4.5. 

over  the  figures;  thus,  146.03817  would  be  146i0  3  8  17.  In 
1640,  the  "Arithmetica  Practica"  of  Adrian  Metius  contains 
sexagesimal  fractions,  but  not  decimal  ones  ;  and  a  work  by 
Joh.  Ilenr.  Alsted,  in  1641,  containing  a  slight  treatise  on 
arithmetic  and  algebra,  says  nothing  about  decimal  fractions. 

About  this  time  the  subject  of  decimals  must  have  been 
pretty  generally  understood;  for  in  "Moore's  Arithmetick," 
1650,  the  subject  of  decimals  is  quite  thoroughly  presented 
and  the  contracted  methods  of  multiplication  and  division  are 
given.  Noah  Bridges,  in  his  "Arithmetick  Natural  and  Deci- 
mal," has  an  appendix  on  decimals,  though  the  author  expresses 
his  disapproval  of  the  use  which  some  would  make  of  decimals, 
averring  that  the  rule  of  practice  is  more  convenient  in  many 
cases.  John  Wallis,  1657,  uses  the  old  decimal  notation  12345, 
but  he  afterwards  adopts  the  usual  point  in  his  algebra;  and 
subsequently  decimals  seem  to  have  been  no  longer  regarded 
as  a  novelty,  but  took  their  place  along  with  the  other  accepted 
subjects  and  methods  of  arithmetic. 

It  may  be  supposed  that  the  publication  of  the  tables  of  log- 
arithms was  necessarily  connected  with  the  knowledge  and  use 
of  decimal  arithmetic  ;  but  this,  Dr.  Peacock  thinks,  is  not  so. 
The  theory  of  absolute  indices,  in  its  general  form  at  least,  was 
at  that  time  unknown  ;  and  logarithms  were  not  considered 
as  the  indices  of  the  base,  but  as  a  measure  of  ratios  merely. 
Under  this  view  of  their  theory,  it  was  a  matter  of  indifference 
whether  we  assumed  the  measure  of  the  ratio  of  10  to  1  to  be 
one,  ten,  a  hundred,  ten  millions,  or  ten  billions,  the  number 
assumed  by  Briggs  in  his  system  of  logarithms.  Thus,  whether 
the  logarithms  are  expressed  by  decimals  or  integers,  they  will 
have  the  same  characteristics,  and  their  use  in  calculation  la 
29 


4:50  THE    PHILOSOPHY    OF   ARITHMETIC. 

exactly  the  same.  It  is  under  the  integral  forms  that  the  loga- 
rithms are  given  in  the  earlier  tables,  such  as  those  of  Napier, 
Briggs,  Kepler,  etc. 

This  statement  will  sufficiently  explain  the  reason  why  no 
notice  is  taken  of  decimals  in  the  elaborate  explanations  which 
are  given  of  the  theory  and  construction  of  logarithms  by  Na- 
pier, Briggs,  and  Kepler ;  and  indeed  we  find  no  mention  of  them 
in  any  English  author  between  1619  and  1631.  In  that  year 
the  Logarilhmical  Arithmetike  was  published  by  Gellibrand, 
a  friend  of  Briggs  who  died  the  year  before,  with  a  much  more 
detailed  and  popular  explanation  of  the  doctrine  of  logarithms 
than  was  to  be  found  in  Briggs's  Arilhmetica  Logarilhmica.  It 
is  there  stated  that  the  logarithms  of  19695,  1969  f^,  19T*$^  are 
respectively  4,29435  etc.,  3,29435  etc.,  1,29435  etc.,  differing 
merely  in  their  characteristic;  and  y5^,  r%9inF'are  called  decimal 
fractions.  Rules  are  also  given  for  the  reduction  of  vulgar 
fractions  to  decimals,  by  a  simple  proportion;  and,  lastly,  a 
table  for  the  reduction  of  shillings,  pence,  and  farthings  to  deci- 
mals of  a  pound  sterling. 

The  Decimal  Point. — The  final  and  greatest  improvement 
in  the  system  of  decimal  arithmetic,  by  which  the  notations  of 
decimals  and  integers  are  assimilated,  was  the  introduction  of 
the  decimal  point,  and  much  labor  has  been  spent  to  ascertain 
its  author.  According  to  Dr.  Peacock,  the  decimal  point  was 
introduced  by  Napier,  the  illustrious  inventor  of  logarithms. 
In  writing  decimals  Napier  seems  to  have  generally  employed 
the  method  of  Stevinus,  which  was  to  indicate  the  decimal 
places  by  primes,  seconds,  etc. ;  but  there  are  at  least  two  in- 
stances in  which  he  used  a  character  as  a  decimal  separatrix. 
The  first  is  an  example  of  division  in  which  he  writes  1993,273, 
using  a  comma,  and  then  presents  his  answer  in  the  form  1993 
2>  jfr  3"/#  ^he  other  instance  occurs  in  a  problem  in  multi- 
plication, in  which  he  draws  a  line  down  through  the  places  of 
the  partial  products  that  would  be  occupied  by  the  decimal 
point;    but  in   the   sum  he   uses  the  exponents   of  Stevinus, 


ORIGIN   OF   DECIMALS.  451 

which  thus  combines  both  methods,  and  stands  1994  |  9'  1" 
6'"  0"". 

The  problems  in  which  these  occur  are  found  in  the  Itabdol- 
ogia,  published   in    1617,  in  which 

he  mentions  the  invention  of  Stevi-  118000 

nus  in  terms  of  highest  praise,  and  \±i 

explains  his  notation  without  notic-  402 

ing   his   own   simplification   of  it.  429 

The   use  of  the   comma,  above  re-  8«1094,000(1993,2T3 

ferred   to,  is  presented    in  the  ac-  3388 

companying  solution,  in  which    it  3888 

is   required    to  divide   861094   by  1296 

432.     I  present  but  a  part  of  the  etc. 

process  of  division.  mi  ..     ,  .    ,ftAOOHO 

F  „,,  „    .,  ,.    ,    7.         The  quotient  is  1993,273, 

The    use    of   the    vertical    line  or  jggg  y  yr  §m 

is  found  in  an  example  of  ab- 
breviated multiplication  which  occurs  in  the  solution  of  the 
following  problem:  "If  31416  be  the  approximate  value  of  the 
circumference  of  a  circle  whose  diameter  is  10,000,  what  is  the 
numerical  value  of  the  circumference  of  a  circle  whose  diame 
ter  is  635?"  This  solution  is  said  to  be  the  first  example 
found  of  this  abbreviated  multiplication  ;  the  use  of  it,  how- 
ever, became  very  popular  in  a  short  time  afterward,  being  es- 
pecially useful  in  the  multiplication  of  the  large  numbers  which 
were  made  use  of  in  the  construction  of  the  tables  of  sines,  etc. 
This  seems  like  a  very  near  approach  to  the  decimal  point, 
if  it  is  not  indeed  the  introduction  of  it ;  but  De  Morgan  main- 
tains that  Napier  only  used  his  comma  or  line  as  a  rest  in  the 
process,  and  not  as  "a  final  and  permanent  indication,  as  well 
as  a  way  of  pointing  out  where  the  integers  end  and  the  frac- 
tions begin."  It  must  be  admitted  that  the  use  of  the  separatrix 
was  merely  incidental,  and  not  the  practice  of  Napier  ;  but  he 
seems  to  be  the  first  to  use  a  mark  for  this  purpose,  even  in- 
cidentally, and  there  can  be  no  doubt  that  even  this  incidental 
use  had  very  great  influence  in  leading  to  the  general  adoptioL 
of  a  decimal  point. 


452  THE   PHILOSOPHY   OF   ARITHMETIC. 

Do  Morgan  thinks  that  Richard  Witt,  who  published  a  work, 
four  years  before  Napier,  "made  a  nearer  approach  to  the  dec- 
imal point"  than  Napier;  yet  he  says,  "I  can  hardly  admit 
him  to  have  arrived  at  the  notation  of  the  decimal  point "  Witt, 
in  a  work  published  in  1613,  presents  some  tables  of  compound 
interest,  in  which  decimal  fractions  are  used.  The  tables  are 
constructed  for  ten  millions  of  pounds,  seven  figures  are  cut 
off,  and  the  reduction  to  shillings  and  pence  with  a  temporary 
decimal  separatrix,  in  the  form  of  a  vertical  line,  is  introduced 
when  wanted,  as  may  be  seen  on  page  446. 

But  though  his  tables  are  distinctly  stated  to  contain  only 
numerators,  the  denominator  of  which  is  always  unity  followed 
by  ciphers,  and  though  he  had  arrived  at  a  complete  and 
permanent  command  of  the  decimal  separator,  and  though  he 
always  multiplies  or  divides  by  a  power  of  10  by  changing  the 
place  of  the  decimal  separator,  which  is  a  vertical  line,  yet 
De  Morgan  thinks  he  gave  no  "meaning  to  the  quantity 
with  its  separator  inserted."  He  thinks  that  if  Witt  had  been 
"asked  what  his  123  |  456  was,  he  would  have  answered:  It 
gives  123tVt)V  not  it  is  las^ftfr" 

Briggs,  the  author  of  the  common  system  of  logarithms,  was 
a  disciple  of  Napier,  and  might  have  been  expected  to  adopt 
Napier's  method  of  writing  decimals.  We  find,  however, 
that  in  1624,  instead  of  using  a  decimal  point  he  draws  a  line 
under  the  decimal  terms,  omitting  the  denominator;  thus, 
5  9321.  A  work  by  Albert  Girard,  published  in  1629  at 
Amsterdam,  is  remarkable  as  using  the  decimal  point  on  a 
single  occasion.  Oughtred,  in  his  Clavis,  published  in  163L, 
uses  both  the  vertical  and  sub-horizontal  separatrix,  thus 
shutting  up  the  numerator  in  a  semi-rectangular  outline,  as 
23|456  for  23.456.  William  Webster's  work,  published  in 
1634,  treats  of  decimals  as  a  thing  generally  known  ;  but  does 
not  make  use  of  the  decimal  point,  using  the  partition  line  to 
separate  integers  and  decimals.  In  1657  John  Wallis  pub- 
lished  a  work  in  which  the  old  notation,  12  345,  was  used^ 


ORIGIN   OF   DECIMALS.  458 

but  he  subsequently  adopted  the  decimal  point  in  his  algebra. 

12  3  4  5 

In  1G43,  the  notation  used  in  Johnson's  arithmetic  is  £3|2  2  9  1  9, 
and  312500,  and  34,G25,  and  sometimes  358149411  fifths.  Kav- 
anagh  says  that  the  present  notation  was,  for  the  first  time, 
clearly  set  forth  in  some  editions  of  Wingate's  arithmetic,  1650. 
On  the  Continent  the  notation  used  was  12  345  or  12[345,  even 
in  works  of  the  highest  repute,  up  to  the  beginning  of  the  18th 
century. 

The    following    summary   presents   some   of   the   different 
methods  of  writing  decimals  which  are  found  among  the  early 
writers  on  arithmetic,  both  in  England  and  on  the  Continent: 
34.  1'.  4".  2'".  6""  34  1426 

0)      (2)     (S)      (4) 

34.  1  .  4  .  2  .  6  34|1426 

34.  1  .  4  .  2  .  6  34*1426 

34.1426""  34,1426 

34.1426<4> 
It  is  believed  that  Gunter,  who  was  born  in  1581,  did  more 
for  the  introduction  of  the  decimal  point  than  any  one  of  his 
cotemporaries.  He  first  adopted  the  notation  of  Briggs,  but 
gradually  dropped  it  and  substituted  the  decimal  point.  In 
one  of  his  works,  De  Morgan  tells  us,  Briggs's  notation  appears 
without  explanation,  and  1 16  04  is  given  as  the  third  proportional 
to  1C0  and  108.  On  a  subsequent  page  a  dot  is  added  to 
Briggs's  notation  in  one  instance;  thus  100/.  in  20  years  at  8 
per  cent,  becomes  466.095/.  At  the  bottom  of  the  same  page, 
Briggs's  notation  disappears  thus:  "It  appeareth  before,  that 
100/.  due  at  the  yeares  end  is  worth  but  92  592  in  ready  money. 
If  it  be  due  at  the  end  of  two  yeares,  the  present  worth  is 
85Z.733;  then  adding  these  two  together,  wee  have  178/.326  for 
the  present  worth  of  100  pound  annuity  for  2  yeares,  and  so 
forward."  After  this  change,  thus  made  without  warning  in 
the  middle  of  a  sentence,  Briggs's  notation  does  not  again  occur 
in  the  part  of  the  work  which  relates  to  numbers.  In  a  pre 
vious  work  on  the  sector,  etc.,  the  simple  point  is  always  used; 


454  THE   PHILOSOPHY    OF   ARITHMETIC. 

but  in  explanation  the  fraction  is  not  thus  written,  but  described 
as  parts.  Thus,  32.81  feet  used  in  the  operation  is,  in  the  de- 
scription of  the  answer,  32  feet  81  parts. 

Fink  says  that  decimal  fractions  were  known  by  Rudolff,  who 
in  the  division  of  integers  by  powers  of  10  cut  off  the  required 
number  of  places  with  a  comma.  He  also  attributes  the  intro- 
duction of  the  decimal  point  to  Kepler,  while  Cantor  says  it  is 
found  in  the  trigonometric  tables  of  Pitiscus,  published  in  1612. 

It  was  some  time  after  this,  however,  before  the  decimal  point 
was  fully  recognized  in  all  its  uses,  even  in  England.  As  long 
as  Oughtred  was  widely  used,  which  was  until  the  end  of  the 
seventeenth  century,  there  must  have  been  a  large  school  of  those 
who  were  trained  to  the  notation  23  I  456.  The  complete  and 
final  victory  of  the  decimal  point  must  be  referred  to  the  first 
quarter  of  the  eighteenth  century. 

It  may  seem  surprising  that  the  decimal  fraction  should  have 
been  introduced  so  late  in  the  history  of  the  science  ;  this  delay, 
however,  admits  of  explanation.  The  decimal  division  of  the 
unit  would  be  of  no  value  until  after  the  Arabic  system  of  notation 
was  adopted.  Even  then  the  introduction  was  necessarily  slow. 
Simple  as  they  now  appear,  the  development  of  decimal  fractions 
was  too  great  an  effort  for  one  mind,  or  even  one  age.  The  idea 
of  their  use  dawned  gradually  upon  the  mind,  and  one  mathe- 
matician taking  up  what  another  had  timidly  begun,  added  an 
idea  or  two,  until  the  subject  was  at  length  fully  conceived  and 
developed. 

The  advantages  of  the  decimal  notation  of  fractions  are  so  ob- 
vious that  they  hardly  need  to  be  specified.  Many  of  the  opera- 
tions upon  fractions  are  thereby  greatly  simplified,  and  others  are 
entirely  avoided.  The  fundamental  operations  of  addition,  sub- 
traction, multiplication  and  division,  are  the  same  as  in  integers, 
and  the  cases  of  reduction  to  lower  terms,  common  denominator, 
etc.,  do  not  occur  at  all.  The  advantages  would  have  have  been 
still  greater  if  the  basis  of  the  numeral  scale  had  been  twelve  in- 
stead of  ten,  as  appears  from  a  previous  discussion. 


CHAPTER  II. 

THE   TREATMENT    OF   DECIMALS. 

A  DECIMAL  FRACTION  is  a  Dumber  of  the  decimal 
divisions  of  a  unit;  or  it  is  a  number  of  tenths,  hundredths, 
etc.  Some  authors  define  it  as  a  fraction  whose  denominator 
is  ten  or  some  power  of  ten ;  and  others  as  a  fraction  whose 
denominator  is  one  followed  by  one  or  more  ciphers.  Both 
of  these  definitions  are  correct,  but  seem  less  satisfactory  than 
the  one  first  presented.  They  are  objectionable  on  account  of 
not  expressing  the  kind  of  fractional  unit,  but  rather  indicating 
its  nature  by  describing  the  denominator  of  the  fraction. 

A  Decimal  Fraction  may  be  expressed  in  two  ways — in  the 
form  of  a  common  fraction,  or  by  means  of  the  decimal  scale. 
When  expressed  by  the  scale  it  is  distinguished  from  the 
general  meaning  of  the  term  decimal  fraction  by  calling  it  a 
Decimal.  A  Decimal  may  thus  be  defined  as  a  decimal 
fraction  expressed  by  the  decimal  method  of  notation.  Thus 
t5u>  iV^  e^c-»  are  decimal  fractions,  but  not  decimals;  while 
.5,  .45,  etc.,  are  both  decimal  fractions  and  decimals.  This 
distinction  is  convenient  in  practice,  and  is  believed  to  be 
strictly  logical.  It  has  not  been  generally  adopted,  but  there 
seems  to  be  a  growing  tendency  towards  such  a  distinction. 
In  popular  language,  however,  we  use  the  term  "  decimal 
fraction"  as  equivalent  to  a  decimal. 

Notation. — The  decimal  fraction,  as  expressed  by  the  decimal 
scale,  has  no  denominator  written,  the  denominator  being 
indicated  by  a  point  before  the  numerator.  This  notation,  aa 
already  seen,  arises  from  that  of  integers,  and  is  merely  an 

(455) 


456  THE    PHILOSOPHY    OF    ARITHMETIC. 

extension  of  it.  Beginning  at  units'  place,  by  a  beautiful 
generalization,  numbers  are  regarded  as  increasing  toward  the 
left  and  decreasing  toward  the  right,  in  a  ten-fold  ratio,  the 
result  of  which  is  a  decimal  division  of  the  unit,  corresponding 
to  each  decimal  multiple  of  it. 

In  order  to  distinguish  between  the  integral  and  fractional 
expression  and  locate  each  term  properly,  a  point  or  separatrix 
is  used.  Various  marks  have  been  employed  for  this  purpose, 
at  different  times,  but  the  period  is  now  generally  adopted. 
The  origin  of  this  use  of  the  decimal  separatrix  is  discussed  in 
the  previous  chapter.  Sir  Isaac  Newton  held  ,that  the  point 
should  be  placed  near  the  top  of  the  figures,  thus,  3*56,  to 
prevent  it  from  being  confounded  with  the  period  used  as  a 
mark  of  punctuation. 

Gases. — The  cases  in  decimals,  it  is  evident,  must  be  nearly 
the  same  as  in  whole  numbers.  The  relation  of  common 
fractions  to  decimals  would,  it  is  natural  to  suppose,  give  rise 
to  one  or  more  new  processes.  A  new  method  of  notation 
having  been  agreed  upon  for  a  special  class  of  common 
fractions,  the  inquiry  naturally  arises, — Can  other  common 
fractions  be  expressed  as  decimals,  and  how?  We  thus  begin 
to  pass  from  common  fractions  to  decimals ;  and,  reversing 
this  process,  pass  back  from  decimals  to  common  fractions. 
This  gives  rise  to  a  process  kuown  as  the  Reduction  of  Fractions, 
embracing  the  two  cases  of  reducing  common  fractions  to  deci- 
mals, and  its  converse,  decimals  to  common  fractions.  The 
reduction  of  common  fractions  to  decimals  gives  rise  to  a  par- 
ticular kind  of  decimals  called  circulates,  which  require  an 
independent  treatment.  The  other  cases  of  decimals  are  the 
same  as  in  whole  numbers. 

Method  of  Treatment. — The  method  of  treating  decimals  is 
quite  similar  to  that  of  whole  numbers.  Indeed,  they  so  closely 
resemble  integers  that  many  authors  have  been  of  the  opinion 
that  they  should  be  presented  with  them.  It  is  claimed  that 
there  is  but  one  principle  in  the  expression  of  integers  and 


TREATMENT    OF    DECIMALS.  457 

decimals,  and  that  the  processes  and  reasoning  are  the  same, 
whether  the  scale  is  ascending  or  descending.  It  is  therefore 
concluded  that  the  notation  of  decimals  should  be  presented 
with  that  of  integers,  and  that  the  fundamental  processes  of 
addition,  subtraction,  etc.,  should  be  applied  to  them  both  in 
the  same  connection. 

There  are,  however,  valid  objections  to  this  seemingly  plausi- 
ble inference  It  will  be  admitted  that  the  mechanical  opera- 
tions are  the  same ;  but  the  reasoning  processes,  in  at  least  two 
of  the  fundamental  operations,  are  not  identical.  The  fixing 
of  the  decimal  point  in  multiplication  and  division,  would  be 
entirely  too  difficult  to  be  presented  along  with  the  fundamental 
operations  of  integers.  Besides,  it  would  be  illogical  to  separate 
one  class  of  fractions  from  the  general  subject  of  fractions  ;  and 
moreover,  one  process,  namely  the  reduction  of  decimals,  could 
not  be  considered  until  after  common  fractions  had  been  dis- 
cussed. These  considerations  have  been  sufficient  to  prevent 
authors  of  arithmetic  from  uniting  the  treatment  of  decimals 
with  that  of  integers,  and  will,  I  doubt  not,  continue  to  sepa- 
rate them. 

Numeration. — In  the  treatment  of  decimals,  the  first  thing 
to  be  considered  is  the  method  of  reading  and  writing  them,  or 
their  Numeration  and  Notation.  These  processes  present  sev- 
eral points  worthy  of  notice,  points  which  seem  to  have  escaped 
the  attention  of  the  writers  on  arithmetic.  Ilaving  introduced 
the  subject  of  decimals  by  explaining  that  the  first  place  to  the 
right  of  units  is  tenths,  the  second  place  hundredths,  etc.,  it  im- 
mediately follows  that  .45  is  read  "4  tenths  and  5  hundredths," 
but  it  does  not  immediately  follow,  as  many  arithmeticians  are 
in  the  habit  of  assuming,  that  it  is  read  "45  hundredths."  If, 
however,  it  is  first  explained  that  T%  is  written  .4,  and  T4^,  .45. 
then  it  does  not  immediately  follow  that  .45  is  read  "4  tenths 
and  5  hundredths."  The  usual  method  of  presenting  decimals 
is  to  explain  that  the  first  place  to  the  right  of  the  decimal  point 
is  tenths,  the  second  place  hundredths,  etc.;  it  should  then  be 
20 


458  THE    PHILOSOPHY    OF    ARITHMETIC. 

shown  that  the  decimal  can  be  otherwise  read.  Thus,  suppose 
we  have  the  decimal  .45:  this  expresses  primarily  4  tenths  and 
5  hundredths  ;  and  since  4  tenths  equals  40  hundredths,  and  40 
hundredths  and  5  hundredths  are  45  hundredths,  the  expression 
45  may  also  be  read  45  hundredths.  This  must  be  explained 
if  we  desire  to  preserve  the  chain  of  logical  thought  in  our 
treatment. 

From  this  it  is  seen  that  in  practice  there  are  two  methods 
of  reading  decimals,  which  may  be  expressed  as  follows : 

1.  Begin  at  the  decimal  point  and  read  in  succession  the 
value  of  each  term  belonging  to  the  decimal,  or 

2.  Read  the  decimal  as  a  whole  number,  and  annex  the  name 
of  the  right-hand  decimal  place. 

It  will  be  noticed  that  in  reading  a  large  decimal  we  should 
numerate  from  the  decimal  point  to  derive  the  denominator, 
and  toward  the  decimal  point  to  determine  the  numerator. 

Notation. — The  writing  of  decimals,  when  conceived  or  read 
to  us,  presents  several  points  of  interest.  When  the  decimal  is 
conceived  analytically,  that  is,  as  so  many  tenths,  hundredths, 
etc.,  we  write  it  by  the  following  rule : 

1.  Fix  the  decimal  point  and  write  each  term,  in  its  proper 
decimal  place. 

If  the  decimal  is  conceived  synthetically,  that  is,  as  a  number 
of  ten-thousandths,  or  a  number  of  millionths,  etc.,  we  write 
it  by  the  following  rule  : 

2.  Write  the  numerator  as  an  integer,  and  then  place  the 
decimal  point  so  that  the  right-hand  term  shall  express  the  de- 
nomination of  the  decimal. 

In  writing  a  decimal  in  which  the  numerator  does  not  occupy 
the  required  number  of  decimal  places,  it  is  not  readily  seen 
where  to  place  the  decimal  point,  and  how  many  ciphers  to  pre- 
fix.    The  best  practical  rule  in  this  case  is  the  following. 

3.  Write  the  numerator  as  an  integer,  and  then  begin  at 
the  right  and  numerate  backward,  filing  vacant  places  with 
ciphers,  until  we  reach  the  required  denomination,  and  to  the 
expression  thus  obtained,  prefix  the  decimal  point. 


TREATMENT    OF    DECIMALS.  459 

Thus,  to  write  475  millionths,  we  first  write  475 ;  then  be- 
ginning at  the  5,  we  numerate  toward  the  left,  saying  tenths, 
hundredths,  thousandths,  ten-thousandths  (writing  a  cipher), 
hundred-thousandths  (writing  a#  cipher),  millionths  (writing  a 
cipher),  and  then  place  the  decimal  point. 

Several  other  methods  have  been  suggested  for  writing 
decimals,  among  which  is  the  following,  by  Prof.  Ilenkle.  It 
is  seen  that  the  tens  of  any  number  of  tenths,  the  hundreds  of 
any  number  of  hundredths,  the  thousands  of  any  number  of 
thousandths,  etc.,  each  fall  in  the  order  of  units  when  the 
decimal  is  expressed.  Thus  56  tenths,  is  5.6,  the  5  tens  falling 
in  units'  place ;  2345  hundredths  is  23.45,  the  3  hundreds  falling 
in  units1  place,  etc.     Hence  the  rule, 

1.  Begin  at  the  left  and  write  the  term  corresponding  to  the 
denominator  of  the  decimal  in  the  place  of  units. 

Reduction. — The  methods  of  treating  the  two  cases  of  reduc- 
tion are  very  simple.  In  reducing  a  common  fraction  to  a 
decimal  fraction,  we  reduce  the  different  terms  of  the  numerator 
to  tenths,  hundredths,  etc.,  and  divide  by  the  denominator.  In 
reducing  a  decimal  to  a  common  fraction,  we  express  the  deci- 
mal in  the  form  of  a  common  fraction,  and  then  reduce  it  to  its 
lowest  terms. 

Fundamental  Operations. — Addition  and  subtraction  are 
treated  exactly  as  in  integers,  the  same  rules  applying  to 
both.  The  mechanical  processes  of  multiplication  and  division 
are  the  same  as  in  whole  numbers ;  the  only  difference  being 
the  placing  of  the  point  in  the  product  and  quotient.  There 
are  two  methods  of  explaining  the  location  of  the  decimal  point 
in  multiplication  and  division,  based  upon  the  different  concep- 
tions of  the  origin  of  the  decimal.  One  locates  the  point  by 
the  principles  of  common  fractions ;  the  other  derives  the 
method  from  the  pure  decimal  conception.  The  latter  is  the 
simpler  and  more  practical  method.  These  two  methods  are 
explained  in  my  works  on  written  arithmetic,  and  need  not  be 
presented  here. 


CHAPTER  III. 

NATURE   OF    CIRCULATES. 

THE  adoption  of  the  method  of  expressing  fractions  by  the 
decimal  scale  opened  up  a  new  avenue  of  thought  in  the 
science  of  numbers.  Decimals  were  treated  without  writing  the 
denominator,  and  common  fractions  were  frequently  thrown 
into  the  decimal  form  and  operated  upon  by  means  of  the  rules 
for  whole  numbers.  The  process  of  changing  common  fractions 
into  the  decimal  scale  led  to  the  discovery  of  an  interesting 
class  of  decimals  called  Circulating  Decimals.  These  new 
forms  soon  attracted  the  attention  and  called  forth  the  ingenuity 
of  mathematicians;  and,  when  investigated,  were  found  to 
possess  some  remarkable  and  interesting  properties. 

Origin. — Circulating  Decimals  have  their  origin  in  the 
reduction  of  common  fractions  to  decimals.  In  making  this 
reduction,  we  annex  ciphers  to  the  numerator,  and  divide  by 
the  denominator.  This  division  sometimes  terminates  with  an 
exact  quotient,  and  sometimes  would  continue  on  without 
ending.  When  it  does  terminate,  the  common  fraction  can  be 
exactly  expressed  in  a  decimal;  when  it  does  not  terminate,  if 
the  division  be  carried  sufficiently  far,  a  figure  or  set  of  figures 
will  begin  to  repeat  in  the  same  order.  Such  a  decimal  is 
called  a  circulating  decimal,  or  simply  a  Circulate. 

It  is  thus  seen  that  Circulates  have  their  origin,  not  in  the 
nature  of  number  itself,  but  in  the  method  of  notation  adopted 
to  express  numbers.  They  are  an  outgrowth  of  the  Arabic 
system  of  notation  and  the  decimal  scale  upon  which  it  is 
based.     If  the  scale  of  this  system  were  duodecimal  instead  of 

(  460  ) 


NATURE   OF   CIRCULATES.  461 

decimal,  the  subject  of  Circulates  would  be  greatly  modified. 
Thus  ^,  £,  ^,  etc.,  which  now  give  circulates,  would  then  give 
finite  decimals;  while  J,  j,  TL,  etc.,  would  give  circulating 
decimals. 

Notation, — The  part  of  the  circulate  which  repeats  is  called 
a.  Repetend  A  Repeteud  is  indicated  by  placing  one  or  two 
periods  or  dots  over  it.  A  repetend  of  one  figure  is  expressed 
by  placing  a  point  over  the  figure  which  repeats ;  thus  .3 
expresses  .333,  etc.  A  repetend  of  more  than  one  figure  is 
expressed  by  placing  a  period  over  the  first  and  the  last  figure; 
thus,  6.345  expresses  6.345345,  etc.  Sometimes  the  first  part 
of  a  decimal  does  not  repeat,  while  the  latter  part  does  repeat. 
Such  a  decimal  is  called  a  mixed  circulate.  The  part  which 
repeats  is  called  the  repeating  part ;  the  part  which  does  not 
repeat  is  called  the  non-repeating  or  finite  part  of  the  circulate. 
Thus  4.536  is  a  mixed  circulate  in  which  5  is  the  finite,  or 
non-repeating  part,  and  36  the  repeating  part. 

In  an  expression  consisting  of  a  whole  number  and  a 
circulate,  if  the  whole  number  contains  terms  similar  to  those 
of  the  repetend,  the  repetend  may  be  indicated  by  placing  one 
of  the  dots  over  a  term  in  the  whole  number.  Thus,  suppose 
we  have  the  circulate  54.234234,  etc. ;  this  is  usually  expressed 
thus,  54.234;  but,  since  the  term  just  before  the  decimal  point 
is  the  same  as  the  last  term  of  the  repetend,  it  may  also  be 
expressed  thus,  54.23.  This  indicates  that  423  repeats;  and. 
expanding  the  expression,  we  have  54.23423  etc.,  which, 
expressed  in  the  ordinary  way,  becomes  54.234.  In  the  same 
way,  6.04  denotes  6.046  ;   20.12  denotes  20.i220. 

The  reading  of  a  repetend  is  a  matter  which  often  puzzles 
young  teachers.  Thus,  in  the  case  of  the  repetend  .3,  since 
the  denominator  is  9,  we  cannot  say  "the  decimal  3  tenths;" 
neither  will  it  answer  to  say  "the  decimal  3  ninths;"  how, 
then,  shall  it  be  read?  The  true  reading  is  "the  circulate 
3  tenths."  Calling  it  a  circulate  distinguishes  it  from  tho 
decimal  fraction  3  tenths,  and  also  indicates  that  it  is  equal  to 
3  ninths. 


462  THE    PHILOSOPHY    OF    ARITHMETIC. 

Again,  how  shall  we  read  .436  ?  It  is  not  sufficiently  explicit 
to  say  "the  mixed  circulate  436  thousandths,"  or  "the  mixed 
circulate  4  tenths  and  36  thousandths,"  since  neither  of  these 
expresses  the  idea  exactly.  The  correct  reading  is,  "the  mixed 
circulate  436  thousandths,  whose  non-repeating  part  is  3  tenths 
and  repeating  part  36  thousandths."  There  may  be  other  read- 
ings equally  correct ;  the  one  suggested  is  given  to  lead  teachers 
to  avoid  the  adoption  of  those  which  are  erroneous. 

Definitions. — A  Circulate  is  a  decimal  in  which  one  or  more 
figures  repeat  in  the  same  order.  A  Repetend  is  the  term  or 
series  of  terms  which  repeat.  This  distinction  between  a  cir- 
culate and  a  repetend  should  be  carefully  noted,  as  it  is  not 
always  clearly  understood.  Circulates  are  Pure  and  Mixed; 
Repetends  are  Perfect  and  Imperfect,  Similar  and  Dissimi- 
lar, and  Complementary.  A  Perfect  Repetend  is  one  which 
contains  as  many  decimal  places,  less  one,  as  there  are  units  in 
the  denominator  of  the  equivalent  common  fraction.  Thus,  \= 
.142857,  and  iV=. 05882352941 17647  are  each  perfect  repe- 
tends. 

Similar  Repetends  are  those  which  begin  and  end  respec- 
tively at  the  same  decimal  places;  as  .427  and  .536.  Dissimi- 
lar Repetends  are  those  which  begin  or  end  at  different  decimal 
places.  Especial  attention  is  called  to  this  definition  of  simi- 
lar repetends,  as  it  is  a  departure  from  the  view  usually  taken 
Repetends  which  begin  at  the  same  place  have  usually  been  re- 
garded as  similar;  and  those  which  end  at  the  same  place, 
conterminous.  It  is  thought,  however,  to  be  much  more  pre- 
cise to  regard  repetends  beginning  and  ending  respectively  at 
the  same  places  as  similar.  Repetends  are  surely  not  quite 
similar  if  they  end  at  different  places ;  to  be  similar  they  should 
both  begin  and  end  at  the  same  place.  This  view  makes  it 
necessary  to  employ  some  other  term  to  indicate  a  similarity 
of  beginning.  There  being  no  word  thus  used,  the  term 
cooriginous,  expressing  a  coorigin,  is  suggested.  Its  appro- 
priateness may  be  seen  by  comparing  it  with  conterminous,  de- 


NATURE    OF   CIRCULATES.  46b 

noting  a  coterminatioQ,  which  has  already   been  adopted  to 
denote  a  similarity  of  endings. 

Cases. — Since  circulates  have  their  origin  in  the  reduction 
of  common  fractions  to  decimals,  it  follows  that  the  first  case  in 
the  treatment  of  circulates  is  Reduction.  The  Reduction  of 
Circulates  embraces  three  distinct  cases :  1.  The  reduction  of 
common  fractions  to  circulates;  2.  The  reduction  of  circulate^ 
to  common  fractions;  3.  The  reduction  of  dissimilar  repeteudi' 
to  similar  repetends.  We  have  also  Addition,  Subtraction, 
Multiplication,  and  Division  of  Circulates.  I  have  also  recently 
introduced  in  my  Higher  Arithmetic  the  Greatest  Common 
Divisor  and  Least  Common  Multiple  of  Circulates,  subjects 
not  heretofore  treated  in  any  arithmetical  work.  The  comparison 
of  circulates  with  common  fractions  gives  rise  to  a  number  of 
interesting  truths,  which  will  be  presented  under  the  head  of 
Principles  of  Circulates. 

Method  of  Treatment. — The  method  of  reducing  common 
fractions  to  circulates  is  the  same  as  that  of  reducing  them  to 
ordinary  decimals.  An  abbreviation,  based  upon  a  principle  of 
repetends,  is  sometimes  employed.  The  method  of  reducing 
circulates  to  common  fractions  differs  considerably  from  the 
method  of  reducing  decimals  to  common  fractions.  In  the 
finite  decimal,  the  denominator  understood  is  1  with  as  many 
ciphers  annexed  as  there  are  places  in  the  decimal ;  in  the 
circulate  the  denominator  of  the  repetend  is  as  many  9's  as 
there  are  places  in  the  repetend.  There  are  three  methods  of 
explaining  this  reduction,  as  will  be  shown  in  the  treatment. 

Circulates  can  be  added,  subtracted,  multiplied,  and  divided, 
by  first  reducing  them  to  common  fractions ;  or  they  may  be 
expanded  sufficiently  far  so  that  the  repeating  figures  may 
appear  in  the  result.  Both  of  these  methods  are  objectionable 
on  account  of  their  length,  and  are  therefore  not  usually 
employed.  In  the  addition  and  subtraction  of  circulates,  it  i 
better  to  reduce  them  to  similar  repetends  and  then  perform 
the  operation.  In  the  multiplication  and  division  of  circulates, 
a  slight  modification  of  this  method  is  employed. 


CHAPTER   IV. 

TREATMENT   OF   CIRCULATES. 

TIIE  Treatment  of  Circulates  embraces  the  operations  of 
Reduction,  Addition,  Subtraction,  Multiplication,  Division, 
Greatest  Common  Divisor,  Least  Common  Multiple,  etc.,  and 
the  Principles  of  Circulates.  Attention  will  be  called  to  the 
treatment  of  several  of  these  subjects,  and  a  distinct  chapter 
will  be  devoted  to  the  Principles  of  Circulates. 

Reduction  of  Circulates. — The  Reduction  of  Circulates  is 
conveniently  treated  under  four  cases: 

1.  To  reduce  common  fractions  to  circulates. 

2.  To  reduce  a  pure  circulate  to  a  common  fraction. 

3.  To  reduce  a  mixed  circulate  to  a  common  fraction. 

4.  To  reduce  dissimilar  repetends  to  similar  ones. 

1.  To  reduce  common  fractions  to  circulates. — The  gen- 
eral method  of  reducing  common  fractions  to  circulates  is  to 
annex  ciphers  to  the  numerator  of  the  common  fraction,  and 
divide  by  the  denominator  continuing  the  division  until  the 
figures  of  the  circulate  begin  to  repeat.  Thus,  to  reduce  -^  to 
a  circulate,  we  annex  ciphers  to  the  numerator  5,  divide  by  the 
denominator  12,  indicate  the  repeating  figure  by  placing  a  period 
over  it ;  and  we  Lave  the  circulate  .41(3. 

When  the  circulate  consists  of  many  figures,  the  process  of 
reduction  may  be  abbreviated  by  employing  some  of  the  prin- 
ciples of  rcpotends.  Thus,  suppose  it  be  required  to  reduce  -^ 
to    a   repetend.     By    actual    division   to    five   places,  we  find 

2V=0.03448^-. 
Now  7^-  is  8  times  -fa,  hence  multiplying  this  by  8  we  have 
^=0.2758G^.     Substituting  this  value  of  -£$  in  the  expression 
for  the  value  of  ^,  and  we  have 

^=0.034482758G&. 
(4G4) 


TREATMENT    OF    CIRCULATES.  465 

This,  multiplied  by  6,  gives  ^=0.206896551 7^;  which,  sub- 
stituted iu  the  second  expression  for  -^  gives 

2V=0.034482758620689655172V 
Multiplying   by    7,   we   get    2^=0.24137931034482758620|| ; 
which,  substituted  in  the  third  expression  for  Jg,  gives 
2^=0.03448275862068965517241379310344827586201^. 

As  the  terms  have  begun  to  repeat,  it  is  unnecessary  to 
continue  the  process  any  further.  It  will  be  seen,  on  examina- 
tion, that  the  repetend  consists  of  28  figures,  or  one  less  than 
the  denominator  of  -fa,  and  therefore   is   a   perfect  repetend. 

2.  To  reduce  a  pure  circulate  to  a  common  fraction.— 
There  are  three  distinct  methods  of  explaining  this  case,  as  has 
already  been  stated.  In  order  to  illustrate  these  methods,  we 
will  solve  the  problem,  Reduce  .45  to  a  common  fraction. 

In  the  first  method,  having  proved  by  actual  division  that 
i=-g>  .01=^,  .001  =  9^-9,  etc.,  we  derive  the  denominator  of 
any  circulate  from  its  relation  to  these  given  circulates.  To 
illustrate,  reduce  the  circulate  .45  to  a  common  fraction.  The 
method  is  as  follows:  Since  .01=^,  as  shown  by  operation. 
actual  division,  .45,  which  is  45  times  .6i,  equals  .QJ—  1 
45  times  ^,  or  £f ,  which,  reduced  to  its  lowest  .45=45—5 
terms,  equals  ^. 

By  the  second  method,  we  multiply  the  circulate  by  1  with 
as  many  ciphers  annexed  as  there  are  places  in  the  repetend, 
which  makes  a  whole  number  of  the  repeating  part  of  the 
circulate.  We  then  subtract  the  two  circulates,  and  have  a 
certain  number  of  times  the  given  circulate  equal  to  the  differ- 
ence, from  which  the  given  circulate  is  readily  found.  We  will 
illustrate  by  the  solution  of  the  same  problem. 

Let  C  represent  the  common  fraction 
which  equals  the  circulate ;  we  will  then  (W^Si!?' 

have  C=.4545  etc. ;  multiplying  by  100     i00C=45!4545  etc". 

to  make  a  whole  number  of  the  repeating       qqp 4^ 

part,  we  have  100  times   the   common  fl— 45  —  5 

fraction  equal  to  45.4545  etc. ;  subtract- 
ing once  the  common  fraction  from   100   times  the   common 
30 


466  THE    PHILOSOPHY    OF    ARITHMETIC. 

fraction,  we   have   99   times   the   common   fraction   equal    to 

45.4545  etc.,  minus  .4545  etc.,  which  equals  45;    hence  once 

the  common  fraction  equals  ff>  or  tt* 

By  the  third  method,  the  repetend  is  regarded  as  an  infinite 

series,  the  ratio  being  a  fraction  whose  numerator  is  1,  and 

denominator    1    with   as   many  ciphers  annexed  as  there  are 

places  in  the  repetend.     The  solution 

is  as  follows :    The  repetend  .45  may       m  .    operation. 

be  regarded  as  an  infinite  series,  T\5¥     .45=T4o^-+  x  o4o5OTT+e*c- 

+xo4o5<nr4-etc.     The  formula   for   the  S=-^=1^h-t9A 

a  1 — r 

Bum  of  an  infinite  series  is  S=- *  =  ±s_=_^_ 

1 r  99    tt* 

Substituting  the  value  of  a=T4o%,  and  r^-j-J^,  we  have  S=^y 


r\,  which  equals  ||,  or 


3.    To   reduce  a  mixed  circulate  to  a  common  fraction. — 
There  are  three  distinct  methods  of  reducing  mixed  circulates 
to  common  fractions,  as  in  the  preceding  case.     To  illustrate 
these  methods  we  will  solve  the  problem, 
Reduce  .3i8  to  a  common  fraction.      By        _   operation. 
the   first   method,  we  reason  thus:     The     -318=^  of  3.18 
mixed  circulate. 3 18 equals -^ of  3. 1.8,  which  _._ ?if  =.?IX 

by  the  preceding  case  equals  -^  of  3-J-|,  or  _  35° _  7 

yV  of  3T2r,  which  equals  TT%,  or  fa  "Tn" TT* 

By  the  second  method,  we  reason  as  follows :   Let  C  repre- 
sent the   common   fraction,  then   we 

u    ii    u  /-(        01010  u.    i  OPERATION. 

shall  have  U— .31818  etc.;  multiply-  q_        31818  etc 

ing  bv  10  to  make  a  whole  number    — 77^ 0  ,010 — — 

f  .  J  .  _  10C=     3.1818  etc. 

01   the    non-repeating   part,  we  have     1000C=318.1818  etc. 
L0  times  the  fraction    equals    3.1818     ~~ §900=315 
etc. ;  multiplying  this  by  100  to  make  c=3j.5—  jjl . 

a  whole  number  of  the  repeating  part, 

wt  have  1000  times  the  fraction  equals  318.1818  etc.;  subtract- 
ing 10  times  the  fraction  from  1000  times  the  fraction,  we  have 
990  times  the  fraction  equals  315,  from  which  we  find  the 
fraction  equals  f-J-f ,  or  fa 


TREATMENT    OF    CIRCULATES.  467 


OPERATION. 
.3J8 

.318" 
3 


In  the  previous  method  we  see  that  we  subtract 
the  finite  part  from  the  entire  circulate,  and  divide 
by  as  many  9's  as  there  are  figures  in  the  repe- 
tend,  with  as  many  ciphers  annexed  as  there  are 
decimal  places  before  the  repetend;  hence,  by 
generalizing  this  into  a  rule,  we  may  perform  the  990— 22" 

operation  as  in  the  margin.     This  is  the  method  preferred  in 
practice. 

This  case  may  also  be  solved  by  regarding  the  repetend  as 
an  infinite  series,  and  finding  its  operation. 

sum  by  geometrical  progression,     ■3i8=-&+Tfrfo+  i(tH  0  0+etu 
and     then     adding     it   to    the  S=rJ|Tr-5-1^=^ 

finite   part.       The    solution    is  ^+^=|^=^. 

presented    in    the    margin,    in 

which  it  is  seen  that  we  regard  t  ^§0  as  the  first  term  of  the 
series,  and  y-^  as  the  rate. 

4.  To  reduce  dissimilar  repetends  to  similar  ones.  To  solve 
this  case  it  is  necessary  to  understand  the  following  principles: 

1.  Any  terminate  decimal  may  be  considered  interminate, 
its  repetend  being  ciphers;  thus,  .45  — .450,  or  .45000,  etc. 

2.  A  simple  repetend  may  be  made  compound  by  repeating 
the  repeating  figure;  thus,  .3=. 33=. 3333,  etc. 

3.  A  compound  repetend  may  be  enlarged  by  moving  the 
right-hand  dot  towards  the  right  over  an  exact  number  of 
periods;  thus,  .245^.24545,  etc. 

4.  Both  dots  of  a  repetend  may  be  moved  the  same  number 
of  places  to  the  right;  thus,  .5378=.53783  or  .537837,  etc., 
for  each  expression  developed  will  give  the  same  result. 

5.  Dissimilar  repetends  may  be  made  cooriginous  by  moving 
both  dots  of  the  repetend  to  the  right  until  they  all  begin  at 
the  same  place. 

6.  Dissimilar  repetends  may  be  made  conterminous  by  mov- 
ing the  right-hand  dots  of  each  repetend  over  an  exact  number 
of  periods  of  each  repetend  until  they  end  at  the  same  place. 

The  method  of  treating  this  case  may  be  illustrated  by  the 


4:68 


THE   PHILOSOPHY    OF   ARITHMETIC. 


OPERATION. 

.45  =,.45454545454545 
.4362  =.43623623623623 
.813694^.81369436943694 


following  example:  Make  .45,  .4362,  and  .813694  similar.  Thtf 
solution  is  as  follows:  To  make 
these  repetends  similar,  they  must 
be  made  to  begin  and  end  at  the 
same  place.  To  do  this,  we  first 
move  the  left-hand  dots  so  that  they 
begin  at  the  same  place,  and  then  move  the  right-hand  dots 
over  an  exact  number  of  periods,  so  that  they  will  end  at  the 
same  place.  Now  the  number  of  places  in  the  periods  are  re- 
spectively 2,  3,  and  4 ;  hence  the  number  of  places  in  the  new 
periods  must  be  a  common  multiple  of  2,  3,  and  4,  which  is  12 ; 
we  therefore  move  the  right-hand  dot  so  that  each  repetend 
shall  contain  12  places. 

Divisor  and  Multiple. — The  Greatest  Common  Divisor  of 
two  or  more  decimals  is  the  greatest  decimal  that  will  exactly 
divide  them.  Such  a  divisor  can  be  found  by  reducing  tho 
decimals  to  common  fractions,  and  applying  the  method  for 
common  fractions  ;  but  it  can  also  be  found  by  keeping  them  in 
the  decimal  form ;  and  the  latter  method  is  generally  less 
tedious  and  more  direct.  To  illustrate  the  method,  let  us  find 
the  greatest  common  divisor  of  .375  and  .423.  We  make  the 
two  circulates  similar,  and  sub- 
tract the  finite  part,  which  re- 


duces them  to  fractions  having 
a  common  denominator.  We 
then  find  the  greatest  common 
divisor  of  their  numerators, 
1638,  which  is  the  numerator 
of  the  greatest  common  divisor, 
the  denominator  being  of  tho 
same  denomination  as  the  simi- 
lar decimals ;  hence  the  greatest 
common  divisor  is  i££ffib,  or 
.0001638. 


OPERATION. 

.3757575  .4234234 


•3757572 

3813264 

55692 
49140 

6552 
6552 


4234230 
3757572 


476658 
501228 

24570 
26208 


1638 

9  9  9  9  9  9  0 : 


1638 
.0001638,  G.  C,  D 


TREATMENT    OF    CIRCULATES. 


469 


The  method,  it  is  seen,  consists  in  reducing  the  decimals  to 
a  common  denominator,  finding  the  greatest  common  divisor  of 
their  numerators,  writing  this  over  the  common  denominator, 
and  reducing  the  resulting  fraction  to  a  decimal. 

The  Least  Common  Multiple  of  two  or  more  decimals  is  the 
least  number  that  will  exactly  contain  each  of  them.  Such  a 
multiple  can  be  found  by  reducing  the  decimals  to  common 
fractions  and  applying  the  method  for  common  fractions;  but 
it  can  also  be  found  by  keeping  them  in  their  decimal  form; 
and  the  latter  method  is  preferred,  as  being  generally  more 
direct  and  less  laborious. 

To  illustrate  the  method,  let  us  find  the  least  common  mul- 
tiple of  .327,  i.Oll  and  .075.  We  reduce  the  circulates  to  frac- 
tions having  a  common 
denominator,  as  in  the 
previous  case.  The 
least  common  multiple 
of  these  numerators  is 
275699700,  which  is 
the  numerator  of  the 
least  common  multiple, 
the  denominator  being 
the  common  denomina- 
tor   of    the    fractions. 


.32727 
3 

32724 


OPERATION. 

1.0*1110 
10 


101100 


.07575 
0 

07575 


3 

4 

25 

101 

27  337  1 

3  x  4 x  25  X  101 X  27  X  337=275699700 
=2757.2727,  L.  C.  M. 
-2757.2 


10908 

33700 

2525 

2727 

8425 

2525 

2727 

337 

101 

Reducing    2^^0<>t>, 
the  least  common  mul- 
tiple, to  whole   numbers   and   decimals,  we   have    2757.2,  the 
least  common  multiple. 

It  will  be  seen  that  the  method  consists  in  reducing  the  dec- 
imals to  a  common  denominator,  finding  the  least  common 
multiple  of  their  numerators,  writing  this  over  the  common 
denominator,  and  reducing  the  resulting  fraction  to  a  decimal, 


CHAPTER  V. 

PRINCIPLES  OF   CIRCULATES. 

THE  investigation  of  the  relation  of  circulate  forms  to  com* 
J-  mon  fractions  has  led  to  the  discovery  of  some  very  inter- 
esting and  remarkable  properties.  These  will  be  considered 
under  the  head  of  Principles  of  Circulates,  and  Complemen- 
tary Repetends.  The  subject  being  rather  briefly  treated  in 
the  text-books,  will  be  presented  here  somewhat  in  detail.  A 
brief  and  simple  explanation  will  be  given  in  connection  with 
each  principle. 

1.  A  common  fraction  whose  denominator  contains  no  other 
prime  factors  than  2  or  5,  can  be  reduced  to  a  simple  decimal. 
For,  since  2  and  5  are  factors  of  10,  if  we  annex  as  many  ciphers 
to  the  numerator  as  there  are  2's  or  5's  in  the  denominator,  the- 
numerator  will  then  be  exactly  divisible  by  the  denominator. 

2.  The  number  of  places  in  the  simple  decimal  to  which  a 
common  fraction  may  be  reduced,  is  equal  to  the  greatest  num- 
ber of  2's  or  5's  in  the  denominator.  For,  to  make  the  numer- 
ator contain  the  denominator,  we  must  annex  a  cipher  for  every 
2  or  5  in  the  denominator,  and  the  number  of  places  in  the 
quotient,  which  is  the  decimal,  will  equal  the  number  of  ciphers 
annexed. 

3.  Every  common  fraction,  in  its  lowest  terms,  whose  denom- 
inator contains  other  prime  factors  than  2  or  5,  will  give  an 
interminate  decimal.  For,  since  2  and  5  are  the  only  factors- 
of  10,  if  the  denominator  contains  other  prime  factors,  the  nu- 
merator with  ciphers  annexed  will  not  exactly  contain  the 
denominator;  hence  the  division  will  not  terminate,  and  the 
result  will  be  an  interminate  decimal. 

(470) 


PRINCIPLES   OF   CIRCULATES. 


471 


4.  Every  common  fraction  which  does  not  give  a  simple 
decimal  gives  a  circulate.  For,  in  reducing  a  common  frac- 
tion to  a  decimal,  there  cannot  be  more  different  remainders 
than  there  are  units  in  the  denominator;  hence,  if  the  division 
be  continued,  a  remainder  must  occur  which  has  already  been 
used,  and  we  shall  thus  have  a  series  of  remainders  and  divi- 
dends like  those  already  used,  therefore  the  terms  of  the  quo- 
tient will  be  repeated 

5.  The  number  of  figures  in  a  repetend  cannot  exceed  the 
number  of  units  in  the  denominator  of  the  common  fraction 
which  produces  it,  less  1.  For,  in  reducing  a  common  fraction 
to  a  decimal,  when  the  number  of  decimal  places  equals  the 
number  of  units  in  the  denominator,  less  1,  all  the  possible 
different  remainders  will  have  been  used,  and  hence  the  divi- 
dends, and  therefore  the  quotients  which  constitute  the  circu- 
late, will  begin  to  repeat.  In  many  cases  the  remainders 
begin  to  repeat  before  we  have  as  many  as  the  denominator 

less  1. 

6.  The  number  of  places  in  a  repetend,  when  the  denominator 
of  the  common  fraction  producing  it  is  a  prime,  is  always  equal 
to  the  number  of  units  in  the  denominator,  less  1,  or  to  some 
factor  of  this  number.  For,  the  repetend  must  end  when  it 
reaches  the  point  where  it  has  as  many  places  less  1  as  there 
are  units  in  the  denominator  of  the  producing  fraction;  hence, 
if  it  ends  before  this,  the  number  of  places  must  be  an  exact 
part  of  the  number  of  places  in  the  denominator  less  1,  that  it 
may  terminate  when  it  has  as  many  places  as  the  denominator 
less  1.     This  is  not  generally  true  when  the  denominator  is 

composite,  as  in  ^-,  -g^,  -g^,  ^9»  etc- 

T.  A  common  fraction  whose  denominator  contains  2  s  or 
Vs  with  other  prime  factors,  will  give  a  mixed  circulate,  and 
the  number  of  places  in  the  non-repeating  part  will  equal  the 
greatest  number  of  2's  or  5's  m  the  denominator.  Dividing 
first  by  the  2's  and  5's,  we  shall  have  a  decimal  numerator 
containing  as  many  places  as  the  greatest  number  of  2's  or  5's 


472  THE    PHILOSOPHY    OF    ARITHMETIC. 

(Priii.  2).  If  we  now  divide  by  the  other  factors,  the  dividends 
consisting  of  the  terms  of  the  decimal  numerator  will  not  give 
the  same  series  of  remainders  as  when  we  have  a  series  of 
dividends  with  ciphers  annexed;  hence  the  circulate  will  begin 
directly  after  the  last  place  of  these  decimal  terms.  To  illustrate, 
take  -g-^Q,  and  factor  the  denominator,  and  we  have 

T5T5"  =  ^ = z z  j 

2x5x5xr 

dividing  by  the  2  and  the  5's  we  have  -^,  in  which  it  is  evident 
the  circulate  must  begin  in  the  third  decimal  place,  just  as  the 
circulate  from  -f-  begins  in  the  first  decimal  place. 

8.  When  the  reciprocal  of  a  prime  number  gives  a  perfect 
repetend,  the  remainder  which  occurs  at  the  close  of  the  period 
is  1.  For,  since  the  reduction  of  the  fraction  to  a  circulate 
commenced  with  a  dividend  of  1  with  one  or  more  ciphers 
annexed,  that  the  quotients  may  repeat  we  must  begin  with 
the  same  dividend,  and  therefore  the  remainder  at  the  close  of 
the  period  must  be  1. 

9.  When  the  reciprocal  of  any  prime  number  is  reduced  to 
a  repetend,  the  remainder  which  occurs  when  the  number  of 
decimal  places  is  one  less  than  the  prime,  is  1.  For,  since  the 
number  of  decimal  places  in  the  period  equals  the  denominator 
less  1,  or  is  a  factor  of  the  denominator  less  1,  at  the  close  of  a 
period  consisting  of  as  many  places  as  the  denominator  less  1, 
there  will  be  an  exact  number  of  repeating  periods,  and  therefore 
che  remainder  will  be  1. 

10.  A  number  consisting  of  as  many  9's  as  there  are  units 
in  any  prime  less  1,  is  divisible  by  that  prime.  For,  if  we 
divide  1  with  ciphers  annexed  by  a  prime,  after  a  number  of 
places  1  less  than  the  prime,  the  remainder  is  1 ;  hence  1  with 
the  same  number  of  ciphers  annexed,  minus  1,  would  be  exactly 
divisible  by  the  prime ;  but  this  remainder  will  be  a  series  of 
9's,  therefore  such  a  series  of  9's  is  divisible  by  the  prime. 
Thus  999999  is  divisible  by  7. 

11    A  number  consisting  of  as  many  Vs  as  there  are  units 


PRINCIPLES   OF    CIRCULATES.  473 

in  any  prime  (except  3),  less  1,  is  divisible  by  that  prime 
For  the  prime  is  a  divisor  of  a  series  of  9's  (Prin.  10),  which 
is  equal  to  9  times  a  series  of  l's;  and  since  9  and  the  prime 
are  relatively  prime,  and  the  prime  is  a  divisor  of  9  times  a 
series  of  l's,  it  must  be  a  divisor  also  of  a  series  of  l's.  Thus 
111111  is  divisible  by  7 ;  also  1111111111  is  divisible  by  11. 

12.  A  number  consisting  of  any  digit  used  as  many  times 
as  there  are  units  in  a  prime  (except  3),  less  1,  is  divisible  by 
that  prime.  For,  since  such  a  series  of  l's  is  divisible  by  the 
prime,  any  number  of  times  such  a  series  of  l's  will  be  divisible 
by  the  prime.  Hence  222222,  333333,  444444,  etc.,  are  each 
divisible  by  7. 

13.  The  same  perfect  repetend  will  express  the  value  of  all 
proper  fractions  having  the  same  prime  denominator,  by 
starting  at  different  places.  Thus,  -}=.  14285714285  etc.  But 
f=. If,  hence  the  part  that  follows  1  in  the  repetend  of  |  is  the 
repetend  off;  that  is,  f=.4285U.  Again,  ^=.14f ;  hence  the 
part  that  follows  .14  in  the  repetend  of  \  is  the  repetend  of  f ; 
that  is,  -§-=. 285714.  In  a  similar  manner  we  find  f=.857142, 
$=.571428  ;  and  the  same  thing  is  generally  true. 

14.  In  reducing  the  reciprocal  of  a  prime  to  a  decimal,  if 
we  obtain  a  remainder  1  less  than  the  prime,  we  have  one- 
half  of  the  repetend,  and  the  remaining  half  can  be  found  by 
subtracting  the  terms  of  the  first  half  respectively  from  9.  Take 
\,  and  let  us  suppose  in  decimating  we  have  reached  a  remain- 
der of  6 ;  now  what  follows  will  be  the  repetend  of  f ,  and  the 
repetend  of  f  added  to  the  repetend  of  \  must  equal  1,  since 
f+^=l;  hence  the  sum  of  these  two  repetends  must  equal 
.999999  etc.,  since  .999999  etc.  equals  1.  Now  in  adding  the 
terms  of  these  two  repetends  together,  that  the  sum  may  be  a 
series  of  9's,  there  must  be  just  as  many  places  before  the  point 
where  6  occurred  as  a  remainder,  as  after;  hence  6  occurred  as 
a  remainder  when  we  were  half  through  the  series. 

Again,  since  the  sum  of  the  terms  of  the  latter  and  the  for- 
mer half  of  the  repetend  equals  a  series  of  9's,  each  term  of 


474  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  first  half  of  the  repetend,  subtracted  from  9,  will  give  the 
corresponding  term  of  the  latter  half  of  the  series. 

All  perfect  repetends  possess  this  property,  and  a  large  num- 
ber of  those  which  are  not  perfect.  Repetends  possessing  this 
property  are  called  complementary  repetends.  The  last  two 
properties  are  of  great  practical  value  in  reducing  common 
fractions  to  repetends. 

15.  Any  prime  is  an  exact  divisor  of  10  raised  to  a  power 
denoted  by  the  number  of  terms  in  the  repetend  of  the  prime, 
less  1 ;  or  of  1 0  raised  toa  power  denoted  by  any  multiple  of  the 
number  of  terms,  less  1.  For,  by  Prin.  6,  the  number  of  places 
in  the  repetend  must  equal  the  number  of  units  in  the  prime,  or 
some  factor  of  that  number ;  hence  the  dividend  used  in  ob- 
taining a  period  must  be  10  raised  to  a  power  equal  to  the 
number  of  terms  in  the  period ;  and  since  the  remainder  at  the 
end  of  the  period  is  1,  the  prime  will  exactly  divide  10  raised 
to  a  power  equal  to  the  number  of  terms  in  the  period,  less  1. 

Both  this  and  principle  6  depend  on  FermaVs  Theorem,  that 
"  pp-1  — 1  is  divisible  by  p  when  p  and  P  are  prime  to  each 
other."  For  10,  the  base  of  the  decimal  system,  is  prime  to 
any  prime  number  except  2  and  5;  hence  lO1*-1  — 1  is  always 
exactly  divisible  by  p,  when  p  is  any  prime  except  2  and  5. 
It  thus  follows  that  in  the  division  of  1  with  ciphers  annexed, 
the  remainder  is  always  1  when  the  number  of  places  in  the 
quotient  is  equal  to  the  number  of  units  in  the  prime.  From 
this  we  can  readily  derive  the  second  part  of  principle  6,  and 
also  principle  15. 

16.  Any  prime  is  an  exact  divisor  of  a  number  when  it  will 
divide  the  sum  of  the  numbers  formed  by  taking  groups  of 
the  number  consisting  of  as  many  terms  as  there  are  figures  in 
the  repetend  of  the  reciprocal  of  that  prime.  We  will  show 
this  for  a  prime  whose  reciprocal  gives  a  repetend  of  three 
places.  The  number  47,685,672,856,  may  be  put  in  the  form 
856  +  672xl03  +  685xl06  +  47xl09,  or  672x  (10s  — 1)4-685 
x(106— l)  +  47x(109  — 1)4-856+672+685  +  47;    but  these 


PRINCIPLES   OF    CIRCULATES.  475 

different  powers  of  10,  diminished  by  1,  are  all  divisible  by  any 
number  whose  reciprocal  gives  a  number  of  three  places,  as  37; 
hence  if  the  sum  of  the  groups,  47  +  685  +  672  +  856,  is  divisible 
by  37,  the  entire  number  is  also  divisible  by  37.  The  same 
may  be  illustrated  with  any  other  number,  and  the  principle  is 
therefore  general.  The  principle  admits,  also,  of  a  general 
demonstration. 

From  this  general  proposition  we  derive  the  following  special 
principles  embraced  under  it: 

1.  Since  the  reciprocals  of  3  and  9  give  a  repetend  of  one 
place,  they  will  divide  a  number  when  they  divide  the  sum  of 
the  digits. 

2.  Since  the  reciprocals  of  11,  33,  and  99,  give  a  repetend  of 
two  places,  they  will  divide  a  number  when  they  divide  the 
sum  of  the  numbers  found  by  taking  groups  of  two  places. 

3.  Since  the  reciprocals  of  27,  37,  and  111,  give  repetends  of 
three  places,  they  will  divide  a  number  when  they  divide  the 
sum  of  the  numbers  formed  by  taking  groups  of  three  places. 

4.  Since  the  reciprocal  of  101  gives  a  repetend  of  four  places, 
it  will  divide  a  number  when  it  divides  the  sum  of  the  numbers 
formed  by  taking  groups  of  four  places. 

5.  Since  the  reciprocals  of  41  and  271  give  repetends  of  five 
places,  they  will  divide  a  number  when  they  divide  the  sum 
of  the  numbers  formed  by  taking  groups  of  five  places. 

6.  Since  the  reciprocals  of  7,  13,  21,  and  39  give  repetends 
of  six  places,  they  will  divide  a  number  when  they  divide  the 
sum  of  the  numbers  formed  by  taking  groups  of  six  places. 


CHAPTER  VI 


COMPLEMENTARY   REPETENDS. 


COMPLEMENTARY  REPETENDS  are  those  in  which  the 
terms  of  the  first  half  of  the  period  are  respectively  equal 
to  9  minus  the  corresponding  terms  of  the  second  half  of  the 
period.  Thus,  in  the  repetend  arising  from  \,  which  is  .142857, 
the  first  term,  1 ,  subtracted  from  9,  leaves  the  fourth  term,  8 ; 
the  second  term  4,  subtracted  from  9,  leaves  the  fifth  term,  5, 
etc.  Complementary  Repetends  include  all  perfect  repetends, 
and  many  repetends  that  are  not  perfect.  From  the  principles 
presented  in  the  preceding  chapter,  the  following  curious  prop- 
erties of  complementary  repetends  will  be  readily  understood : 

1.  If  the  last  half  of  the  terms  of  a  perfect  repetend 
be  written  in  order  under  the  first   half  and 

added  to   the  terms  in  the  first  half  the  sum     S^^', 
will  be  a  succession  of  9's.     Thus,  the  fraction     95652173913 
2^=0434782608695652173913;   and   this  repe-     99999999999 
tend,  written  and  added  as  suggested,  will  give 
a  series  of  9's,  as  is  seen  in  the  margin. 

2.  If  the  remainders  obtained  in  reducing  the  common  frac- 
tion to  a  repetend  be  written  in  the  same  way  and  added,  eacl 
sum  will  be  the  denominator  of  the  common  fraction.  Thus, 
the  remainders  in  reducing  -^  are 

10,     8,   11,   18,   19,     6,"  14,     2,  20,   16,  22, 
13,  15,  12,     5,     4,  17,     9,  21,     3,     7,     1,  which,  added, 
give  23,  23,  23,  23,  23,  23,  23,  23,  23,  23,  23. 

3.  If  we  subtract  the  unit  term  of  the  denominator  of  the 
common  fraction  from  10,  and  multiply  any  term  of  the  repe- 

(476) 


COMPLEMENTARY    REPETENDS.  477 

tend  by  the  remainder,  the  unit  term  of  the  product  will  be  the 
unit  term  of  the  corresponding  remainder.     Thus,  in  -£%, 

.0,  4,  3,  4,  7,  8,  2,  etc.,  are  the  terms  of  the  repetend. 
10-  3  =  7 

0,  8,  1,  8,  9,  6,  4,  etc.,  unit  terms  of  products  and 
remainders. 

4.  A  complementary  repetend,  by  beginning  at  different 
points,  will  be  the  repetend  of  all  proper  fractions  having  the 
same  denominator  as  the  fraction  which  produced  it.  Thus, 
^3=643478208  etc. ;  and  ££=.43478  etc.,  which  begins  with 
the  second  figure  of  the  circulate  of  ^3-.  Again,  -^3-=  34782 
etc.,  which  begins  with  the  third  figure  of  the  circulate  equal  to 
A  etc. 

5.  The  numerator  of  the  fraction  equal  to  any  one  of  the 
several  repetends  beginning  with  the  successive  figures  of  a 
complementary  repetend,  is  the  remainder  left  when  the  pre- 
ceding figure  of  the  repetend  was  obtained.  Thus,  in  the  cir- 
culate of  ^3-,  when  the  first  4  of  the  circulate  was  obtained, 
8  was  the  remainder,  and  8  is  the  numerator  of  the  fraction 
equal  to  the  circulate  .34782  etc. 

The  following  are  all  the  perfect  repetends  whose  denomina- 
tors are  less  than  100: 


^=.0588235294117647 
^=.052631578947368421 
^=.0434782608695652173913 
^=0344827586206896551724137931 
^=0212765957446808510638297872340425531914893617 
!  _  (.016949152542372881355932203389830508474576271186 

inr~~  X  440677966i 
1  =  (.016393442622950819672131147540983606557377049180 

*rr_  X  327868852459 

!  =  (.010309278350515463917525773195876288659793814432 
"      1  989690721649484536082474226804123711340206185567 


478  THE   PHILOSOPHY   OF    ARITHMETIC. 

The  following  repetends  are  complementary,  as  may  be 
seen  by  the  test  for  complementary  repetends,  but  are  not 
perfect : 


^==.076923 
TV=.01369863 


^=.0112359550561797752808  j]?al.f  J 

(Period. 

^=•0099 

T^3=  00970873786407766  |p^f0J 


The  following  common  fractions  give  an  even  number  of 
figures  in  a  period,  but  the  repetends  are  not  complementary, 
as  will  be  seen  by  applying  the  test  for  complementary 
repetends : 

^=.047619 


A=-03 


^=.0204081  etc.,  to  42  places. 

^=0196078431372549. 
A=-025641 

The  following  are  not  complementary  repetends,  but  the 
number  of  places  in  each  repetend  is  an  exact  part  of  the 
number  of  units  less  one  in  the  denominator  of  the  common 
fraction  producing  it: 

^.=  032258064516129 

-^=.02439 

^=.023255813953488372093 

-^=.0188679245283 

-gV=  014925343134328358208955223880597 

^=.01408450704225352112676056338028169 

^=6l2658227848i 

-^=01204819277108433734939759036144578313253 

j   =  (.00934579439252336448598130841121495327102803738 
1TT      (317757 

Professor  Perkins  has  ingeniously  illustrated  some  of  the 
properties  of  repetends,  by  arranging  the  terms  of  a  repetend 
and  the  corresponding  remainders  which  arise  in  obtaining 
them,  in  concentric  circles. 


COMPLEMENTARY    REPETENDS. 


479 


The  following  illustration  and  discussion  I  have  taken  from 
his  work  on  Higher  Arithmetic  published  in  1844: 


In  this  figure,  the  inner  circle  of  figures,  commencing  at  the 
0,  directly  under  the  asterisk,  and  counting  towards  the  right 
hand,  is  the  circulating  period  of  -£§. 

The  outer  circle  of  figures,  commencing  at  the  same  place 
and  counting  in  the  same  direction,  are  the  successive  remain- 
ders which  will  occur  in  the  operation  of  decimating  -£§. 

In  this  circle  of  remainders,  all  the  numbers  from  1  to  28, 
inclusive,  occur,  but  not  in  numerical  order. 

From  what  has  been  previously  explained,  we  infer  the  fol- 
lowing properties,  which  are  common  to  all  perfect  repetends : 

1.  The  sum  of  any  two  diametrically  opposite  terms  of  the 
circle  of  decimals  will  be  9. 

2  The  sum  of  any  two  diametrically  opposite  terms  in  the 
circle  of  remainders  will  equal  the  denominator  29. 

3.  If  we  subtract  the  right-hand  term  of  the  denominator 
from  10,  and  multiply  the  remainder  by  any  decimal  term  of 


480  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  inner  circle,  the  right-hand  figure  of  the  product  will  be 
the  same  as  the  right-hand  figure  of  the  corresponding  re- 
mainder of  the  outer  circle. 

4.  Commencing  the  circle  of  decimals  at  any  point  and 
counting  completely  round,  we  shall  have  the  perfect  repetend 
of  the  common  fraction  whose  denominator  is  the  same  as  in 
the  first  case,  but  whose  numerator  is  the  remainder  in  the 
outer  circle  standing  one  place  to  the  left. 

5.  If  we  divide  the  product  of  any  two  remainders  by  29, 
what  remains  will  be  the  remainder  in  the  outer  circle,  corre~ 
sponding  with  the  place  denoted  by  the  sum  of  the  places  of  the 
two  numbers. 

From  the  fourth  property  it  follows  that  this  same  circle  oi 
decimals  expresses  the  decimal  value  of  all  proper  common  frac- 
tions whose  denominators  are  29. 

A  similar  figure,  formed  from  the  perfect  repetend  of  the 
common  fraction  ^,  possesses  properties  the  same  as  those  just 
explained.  Similar  circles  may  be  formed  for  all  perfect  repe- 
tends. 

If  we  arrange  the  complementary  repetend  arising  from  the 
fraction  y^-  in  the  form  of  a  circle,  as  was  done  for  perfect 
repetends,  it  will  be  seen  that  a  complementary  repetend 
possesses  all  the  properties  ascribed  to  the  perfect  repetend  on 
pages  479  and  480,  except  the  fourth. 


CHAPTER  VII. 

A   NEW   CIRCULATE   FORM. 

IN  The  Normal  Written  Arithmetic,  I  presented  two  or  three 
curious  circulate  forms  which  have  given  rise  to  so  much 
discussion  among  teachers,  that  it  is  thought  well  to  call  at- 
tention to  them  in  this  work.  These  forms  are  ,0J  and  .OjOJ, 
both  of  which  are  intended  to  represent  pure  circulates.  Sim- 
ilar forms  may  also  occur  in  mixed  circulates,  as  .O^OJ  and 
.0^0  JO  £.  Two  questions  have  been  raised  with  regard  to 
these  expressions:  first,  what  do  they  mean?  and  second,  are 
they  legitimate  arithmetical  expressions  ?  I  propose  to  say  a 
few  words  on  their  meaning,  their  origin,  and  their  value. 

Before  discussing  this  circulate  form,  attention  is  called  to 
the  signification  of  the  decimal  expression  .2^.  In  this  expres- 
sion, does  the  J  express  tenths  or  hundredths?  Is  ^  regarded 
as  occupying  one  of  the  places  in  the  decimal  scale,  or  is  it  a 
part  of  the  tenth  ?  It  has  been  held  that  the  £  occupies  hun- 
dredths place ;  but  a  very  slight  consideration  is  sufficient  to  lead 
oue  to  see  that  the  ^  is  one-half  of  a  tenth,  and  not  one-half  of 
a  hundredth.  In  integers  and  fractions,  the  fraction  always 
denotes  a  part  of  a  unit  of  the  term  to  the  right  of  which  it 
stands.  Thus,  in  2J,  the  2  expresses  units,  and  the  ^  is  -J  of  a 
unit.  Now,  if  we  divide  2J  by  10,  we  have  .2J;  in  which  the 
.2  expresses  tenths,  and  by  the  principle,  as  before,  the  ^  would 
be  £  of  a  tenth.  Conversely,  if  we  take  the  expression  .2 §•  and 
multiply  it  by  10,  we  have  .2^xl0=2j,  and  not  2  units  and 
^  of  a  tenth.  The  same  result  will  be  reached  if,  regarding 
the  ^  as  ^  of  a  tenth,  we  reduce  it  to  5  hundredths,  in  which 
31  (481) 


482  THE    PHILOSOPHY    OF    ARITHMETIC. 

case  the  expression  becomes. 25;  for  multiplying  this  by  10,  we 
have  .25  x  10=2.5,  or  2J.  Now,  if  the  above  is  true  for  .2J,  it 
is  equally  true  for  .1^-  or  .0J;  from  which  we  see  that  in  the  ex- 
pression .0J,  the  J  belongs  to  tenths  place,  and  should  not  be 
regarded  as  occupying  hundredths  place. 

These  expressions  may  actually  arise  in  an  investigation. 
Suppose  we  are  required  to  subtract  $12.62  from  $25.62^;  the 
difference  is  evidently  $13.00j;  also  3.4  subtracted  from  6.4| 
equals  3.0 J.  In  the  former  case  it  is  clear  that  the  ^  is  a  half 
of  a  hundredth ;  in  the  latter,  a  half  of  a  tenth.  It  hardly  seems 
necessary,  after  this,  to  say  that  3.J  does  not  express  3  and  | 
of  a  tenth,  as  some  have  supposed ;  and  also  that  the  expres- 
sion is  an  illegitimate  one,  without  any  other  meaning  than 
3J.00.     Let  us  now  consider  the  forms  mentioned  above. 

Origin. — These  circulate  forms  originated  from  an  extension 
of  a  mixed  integer  to  a  mixed  decimal,  and  then  a  further  ex- 
tension of  the  decimal  to  a  circulate.     They  may,  however,  bo 
immediately  derived  from  the  reduction  of  a  common  fraction 
to  a  decimal.     To  illustrate,  reduce  Jg  to  a  decimal.     Follow- 
ing the  ordinary  rule,  we  annex  zeros  to  operation. 
the  numerator  and  divide  by  the  denom-     18)1.00(.0J0J,  etc., 
inator:  18  is  contained  in  10  tenths  J  of  9       or  .0J 
a  tenth  time,  with   1  tenth  or  10  hun-  10 

dredths  as  a  remainder  ;  18  is  contained  ?. 

in  10  hundredths  J  of  a  hundredth  time,  * 

with  1  hundredth  remaining,  etc.  Here  we  observe  that  the  re 
mainders  repeat ;  hence  the  quotient  figures  will  repeat,  and 
Jg-  may  be  regarded  as  equal  to  .OjOjO^  etc.,  or  .0J. 

Again,  let  us  reduce  ^r  to  a  circulate. 

Annex  zeros  to  the  numerator  and  divide  by         operation. 

the  denominator:    29T  is  contained  in  160     29T)16.00(.6i0j 

tenths  J  of  a  tenth  time,  with  115  hundredths  14  85 

as  a  remainder  ;  29?  is  contained  in  115  hun-  115 

99 
dredths  \  of  a  hundredth  time,  with  16  hun-  — - 

1  A 

dredths  as  a   remainder.      The   remainders 


60 

100 
90 

10 


A    NEW    CIRCULATE   FORM.  483 

begin  to  repeat;  hence  the  quotient  figures  will  repeat  and  we 

have  3&V=.6iOi. 

Again,   reduce  -^  to   a   circulate.     We 
divide  the  numerator  by  the  denominator:       operation. 
180   is  contained  in   10  tenths  £  of  a  tenth      180)t.00(.0£0J 
time,  with  a  remainder  of  10  tenths  or  100 
hundredths;    180  is  contained  in  100   hun- 
dredths ^  of  a  hundredth  time,  with  a  re- 
mainder of  10  hundredths.    Here  we  observe 
that  the  last  remainder  is  the  same  as  the  preceding;  hence  we 
conclude  that  i  will  repeat,  and  we  have  yiiF^OsOi- 

Meaning. — Now  what  does  .o|  signify?  First,  it  is  said 
since  J  of  a  tenth  equals  5  hundredths,  that  5  is  the  repetend, 
and  that  the  repeating  term  first  appears  in  the  order  of  hun- 
dredths; or,  in  other  words,  that  .0J  is  the  same  as  the  mixed 
circulate  .05.  This  view  is  founded  on  a  wrong  conception  of 
the  relation  of  a  fraction  to  the  term  on  the  left  of  it,  as  may 
be  seen  from  what  has  already  been  explained,  and  also  shows 
a  wrong  conception  of  a  repetend.  A  repetend  is  simply  the 
term  or  terms  which  repeat,  and  does  not  concern  itself  with 
their  place  or  order  value.  Thus  the  expression  .3  does  not 
mean  that  3  tenths  repeats,  but  simply  that  the  term  3  repeats ; 
hence  .0J  does  not  mean  that  ^  of  a  tenth,  or  5  hundredths  re- 
peats, but  simply  the  expression  0^.  It  is  true  that  .0^-  equals 
.05 ;  but  it  does  not  mean  precisely  the  same  thing,  neither  is 
anything  gained  in  general  by  reducing  it  to  that  form.  This 
will  appear  in  some  of  the  subsequent  expressions. 

Second,  it  has  been  said  that  the  expression  .0J  represents 
an  absurdity.  This  cannot  be,  unless  the  fraction  occurring  in 
a  decimal  place,  with  no  repeating  part,  is  also  absurd.  It  will 
be  admitted  that  .2J  is  a  legitimate  decimal  expression  ;  and, 
if  this  is  so,  then  it  is  conceivable  that  such  an  expression  may 
be  repeated.  This  conception  certainly  involves  nothing  ab- 
surd;  and  hence  the  expression  of  this  idea,  or  .2J2J2J  etc., 
seems  to  be  entirely  legitimate.     This  gives  rise  to  the  expres- 


484  THE    PHILOSOPHY   OF    ARITHMETIC. 

sion  .2i;  and  if  the  form  .2|  is  legitimate,  so  also  is  .0^.  The 
expression  .O^O^O^  etc.,  is  a  legitimate  conception  ;  and  hence 
the  abbreviated  expression  of  it,  .0^,  cannot  represent  an  ab- 
surdity. 

But,  admitting  the  correctness  of  the  conception,  does  one  point 
express  the  idea  that  the  entire  expression  .0^  repeats?  It  is 
clear  that  it  will  not  do  to  use  two  dots,  since  that  would  in- 
dicate that  there  were  two  orders  of  the  decimal  scale  occupied, 
which  cannot  be  unless  such  expressions  as  2 J,  4^,  are  regarded 
as  occupying  tens  and  units  place,  which  no  one  would  claim. 
It  may  be  asked,  however,  whether  the  point  should  be  placed 
over  the  0  or  the  ^ ;  thus,  .6^  or  .0  j,  and  this  it  is  difficult  to 
decide.  Both  characters  occupy  tenths  place,  so  that  it  would 
seem  to  indicate  the  same  thing  when  placed  over  either  char- 
acter. There  is  one  consideration  which  inclines  one  to  place 
the  point  over  the  fraction  If  we  place  it  over  the  0,  it  might 
be  understood  that  the  0  repeats  and  not  the  J;  but  if  placed 
over  the  -J,  it  must  indicate  the  repetition  of  both,  since  the  J 
cannot  be  repeated  without  using  the  0  in  order  to  locate  it. 
To  prevent  ambiguity,  it  would  probably  be  better  to  place  it 
over  both,  or  rather  partly  between  them;  thus,  .0£.  In  the 
case  of  expressions  occupying  more  than  one  place,  the  mean- 
ing will  be  apparent  if  the  point  is  placed  over  the  beginning 
and  end  of  the  expression ;  thus,  .O^O^. 

Value. — These   forms   can   be  readily  reduced  to  common 
fractions,  and  their  value  thus  expressed. 
A  few  examples  will  illustrate  the  method.         0PpRA^ °*' 
Thus,  reduce  .0J  to  a  common  fraction.        iqF— i  O-^etc0" 

Let  F  represent  the  fraction  equal  to  .0^.  qp_1 

Multiplying  the  equation  by  10,  we  have  ~£ 

10F=^  of  a   unit   and  the  circulate   .0j.  F=<j=A* 

Subtracting  the   first  equation   from   the 

second,  we  have  9F=-J,  whence  F=|,  or^. 


OPERATION. 


A   NEW   CIRCULATE   FORM.  485 

Again,  reduce  .OjO^  to  a  common  fraction.     Let  F  represent 
the  fraction  equal  to  .0|0£.     Multiply- 
ing by  100,  we  have  100F  equal  to  J 
of  a  ten,  i  of  a  unit,  and  the  circulate     100f  Zigffi  *£ 
.0J0^;   subtracting  the  first  equation  * 

from  the  second,  we  have  99F   equal  ~~  qJq? 

to  i  of  a  ten  and  ^  of  a  unit,  whence  "^=_99~  X  "6"=A% 

F   equals  ^  of  a  ten  and  ^  of  a  unit  __  ie 

divided  by  99;    multiplying  both  nu- 
merator and  denominator  by  6,  we  have  -^,  since  6  times  ^ 
of  a  unit  equals  2  units,  and  6  times  ^  of  a  ten  equals  3  tens. 

Again,  reduce  .O-JO^O-J-  to  a  common  fraction.    Let  F=.0J0£0 £. 
Multiplying  by    10,  we   have 

inn  i  -j.      1 1      *  -,  j  OPERATION. 

10F  equal  to  \  of  a  unit,  and  ninini 

•O'iOi;   multiplying   again   by         ^pZ^xfJ 
100  ,  we  have  1000F  equal  to         iujj_v2.u3u^ 

i  of  a  hundred,  *  of  a  ten,  J  1000*^0^.0^ 

of  a   unit,    and   the   circulate  990F=0i0£0J— O^OJOS^ 
subtracting   10F   from  ~F=  2     ^  x  -|ft— j^Vs 


|  of  a  hun- 


990 


dred,  £  of  a  ten,  £  of  a  unit,  minus  i  of  a  unit,  or  reducing  the 
^  of  a  ten  to  units,  and  subtracting  the  -J  of  a  unit,  we  have 
990F  equal  to  -J-  of  a  hundred  and  3-^j-  units ;  whence   F= 

Q1Q3   1 

Y    J7  ;  multiplying  both  terms  of  the  fraction  by  30,  we  have 

Vow 

2W0V  since  30  times  3^  equals  91,  and  30  times  £  of  a  hun 
dred  equals  15  hundred. 

These  forms  are  more  curious  than  practical,  and  were  pre- 
sented at  first  merely  as  puzzles.  The  discussion  here  given 
grew  out  of  the  fact  that  their  correctness  has  been  questioned. 


Part  V. 
DENOMINATE  NUMBERS. 


J.  Nature  of  Denominate  Numbers. 


II.  Measures  of  Extension. 


III.  Measures  of  Weight. 


IV.  Measures  of  Value. 


V.  Measures  of  Time. 


VI.  The  Metric  System. 


CHAPTER  I. 

NATURE   OP   DENOMINATE   NUMBERS. 

THE  numerical  idea  is  practically  the  product  of  two  factors 
— Mind  and  Matter.  We  begin  by  numbering  objects,  or 
with  concrete  numbers ;  we  then  withdraw  the  numerical  idea 
from  these  objects,  and  obtain  pure  or  abstract  numbers.  The 
child's  first  numbers  are  invariably  concrete;  from  these  it 
passes  to  number  in  the  abstract.  Subsequently,  in  the  effort 
to  apply  the  numerical  idea  to  quantities  not  existing  in  indi- 
vidual forms,  there  arises  a  third  class  of  numbers  called  De- 
nominate Numbers.  The  nature  of  these  numbers,  which 
seems  not  to  have  been  very  clearly  apprehended  by  arithmeti- 
cians, will  be  discussed  somewhat  in  detail. 

Nature,  regarded  as  how  many  and  how  much,  gives  rise  to 
two  kinds  of  quantity — quantity  of  magnitude  and  quantity  of 
multitude,  called  also  discrete  and  continuous  quantity.  These 
two  classes  of  quantity  are  entirely  distinct,  as  may  be  seen  by 
the  manner  in  which  they  are  primarily  estimated.  Discrete 
quantity  is  immediately  estimated  as  how  many ;  continuous 
quantity  is  primarily  estimated  as  how  much.  Thus,  we  say 
how  many  apples,  how  many  trees,  how  many  birds,  etc.,  not  how 
much  apples,  how  much  trees,  how  much  birds,  etc.  On  the  other 
hand,  we  say  how  much  money,  how  much  land,  how  much  time, 
how  much  do  you  weigh,  etc.;  not  how  many  money,  how  many 
land,  etc.  When,  however,  we  fix  upon  some  unit  of  measure, 
these  latter  quantities  can  also  be  expressed  numerically ;  thus, 
we  may  s&yhow  many  dollars,  how  many  acres,  how  many  pounds 
do  you  weigh,  etc.  In  this  manner,  quantity  of  magnitude,  the 
21*  (  489 ) 


490  THE    PHILOSOPHY   OF    ARITHMETIC. 

how  much,  becomes  definitely  apprehended  as  the  how  many 
We  fix  upon  a  unit  of  measure,  and  estimate  continuous  quantity 
by  comparing  it  with  this  unit  as  a  standard.  This  comparison 
leads  to  a  numerical  apprehension  of  the  quantity  considered; 
we  speak  of  it  as  so  many  of  the  unit  of  measure.  Quantity 
of  magnitude  is  thus  estimated  as  quantity  of  multitude ;  the 
how  much  is  reduced  to  and  regarded  as  the  how  many.  In 
this  manner  arises  a  distinct  class  of  numbers  called  Denomi- 
nate Numbers. 

A  Denominate  Number  is  thus  seen  to  be  a  numerical 
expression  of  quantity  of  magnitude.  Quantity  of  multitude 
exists  primarily  as  units;  in  quantity  of  magnitude  we  fix 
upon  some  particular  portion  of  the  quantity  as  a  unit  of 
measure,  and  estimate  the  quantity  by  the  number  of  times  it 
contains  this  unit.  From  this  conception  of  the  origin  and 
nature  of  a  denominate  number,  we  are  prepared  to  give  a 
scientific  definition  of  it. 

Definition. — A  Denominate  Number  is  a  numerical  expres- 
sion of  quantity  of  magnitude  ;  or,  A  Denominate  Number  is  a 
numerical  expression  of  continuous  quantity;  or,  A  Denom- 
inate Number  is  a  number  of  units  of  quantity  of  magnitude ; 
or,  A  Denominate  Number  is  a  number  in  which  the  unit  is  a 
measure  of  a  quantity  of  magnitude ;  or,  A  Denominate  Num- 
ber is  a  number  in  which  the  unit  is  a  measure.  In  using  the 
last  definition,  we  would  of  course  attach  a  special  meaning  to 
the  term  measure.  It  may  also  be  remarked  that  in  each  defi- 
nition the  expression  continuous  quantity  may  be  substituted 
for  quantity  of  magnitude. 

The  quantities  thus  considered  are  Time,  Weight,  Value,, 
Length,  etc.,  which  are  called  quantities  of  magnitude.  The 
term  magnitude,  meaning  literally  size,  extent,  was  primarily 
applied  to  quantities  occupying  space ;  that  is,  to  something 
possessing  length,  breadth,  and  thickness.  In  this  primary 
sense  it  does  not  include  weight,  value,  and  time,  since  these 
have  no  size  or  extent — no  length,  breadth,  or  thickness.     The 


NATURE    OF    DENOMINATE   NUMBERS.  491 

signification  of  the  term,  however,  has  become  gradually  en- 
larged, until  it  now  includes  every  kind  of  continuous  quantity. 
It  was  used  in  this  sense,  I  think,  as  early  as  the  time  of 
Euclid. 

Unit  of  Measure. — Quantity  of  magnitude  is  numerically 
expressed,  as  already  stated,  by  comparing  it  with  some 
fixed  quantity  of  the  same  species  used  as  a  standard  of  com- 
parison. This  standard  of  comparison  is  called  the  Unit  of 
Measure.  The  Unit  of  Measure  is  a  definite  portion  of  the 
particular  kind  of  quantity  considered.  Thus,  in  weight  we 
take  some  definite  weight  as  the  unit,  and  estimate  the  entire 
weight  in  numbers  by  its  relation  to  the  unit.  The  same  is 
done  in  time,  value,  length,  etc.  The  unit  of  measure  thus 
becomes  the  basis  of  all  these  quantities,  the  quantities  them- 
selves being  definitely  conceived  only  as  we  have  a  definite 
idea  of  the  unit. 

The  units  of  measure  by  which  these  quantities  are  estimated 
do  not  exist  in  nature  ;  they  are  agreed  upon  by  mankind,  and 
are  therefore  artificial.  They  constitute  a  distinct  class  of 
concrete  units,  and  give  rise  to  a  distinct  class  of  Concrete 
Numbers.     It  is  thus  seen  that  there  are  two  classes  of  concrete 

numbers, one  in  which  the  unit  is  natural,  and  another  in 

which  it  is  artificial.  Thus  in  4  trees,  the  unit  tree  is  found 
in  nature,  and  is  therefore  a  natural  unit ;  in  4  pounds,  the  unit 
pound  is  not  found  in  nature,  but  is  fixed  by  man,  and  is  there- 
fore artificial.  This  consideration  leads  us  to  another  definition 
of  denominate  numbers ;  thus, — A  Denominate  Number  is  a 
concrete  number  in  which  the  unit  is  artificial. 

The  terms  natural  and  artificial,  as  here  used,  refer  to  the 
quantities  regarded  as  units  and  not  merely  as  objects.  Thus, 
in  the  number  4  knives,  the  knife,  as  an  object,  is  not  found  in 
nature ;  it  is  a  work  of  art  and  therefore  artificial.  But  though 
artificial  as  an  object,  as  a  unit  of  measure  it  is  natural.  It  is, 
therefore,  entirely  correct  to  speak  of  artificial  and  natural 
units,  and  consequently  to  define  a  denominate  number  as  a 


492  THE   PHILOSOPHY    OF    ARITHMETIC. 

number  of  artificial  units.  The  definition  previously  given, 
however,  is  regarded  as  better. 

Quantities. — The  quantities  of  magnitude,  or  continuous 
quantities,  which  give  rise  to  denominate  numbers,  are  of  sev- 
eral different  kinds.  A  scientific  classification  of  these  quanti- 
ties is  as  follows;  1.  Yalue;  2.  Weight;  3.  Space;  4.  Time. 
The  term  extension  is  more  frequently  used  than  space;  we 
generally  speak  of  measures  of  extension  rather  than  measures 
of  space.  Space  includes  several  distinct  forms  of  extension; 
length,  surface,  volume,  etc.;  and  hence  a  popular  classification, 
and  one  a  little  more  convenient  in  practice,  is  the  following: 
1.  Yalue;  2.  Weight;  3.  Length;  4.  Surface;  5.  Volume;  6. 
Capacity;  7.  Angles;  8.  Time. 

The  unscientific  manner  in  which  the  subject  of  denominate 
numbers  has  been  presented,  has  led  to  many  incorrect  ideas 
concerning  them.  Some  writers  have  considered  them  under 
the  head  of  "Weights  and  Measures;"  seeming  not  to  know 
that  the  measure  of  the  force  of  gravity  is  just  as  much  a 
measure  as  the  measure  of  length.  One  writer  says  they  may 
be  divided  into  three  general  classes — Currency,  Measure  and 
Weight — seeming  not  to  understand  that  currency  is  a  measure 
of  value,  and  weight  a  measure  of  gravity.  It  is  hoped  that 
the  views  here  presented  will  lead  arithmeticians  and  teachers 
to  a  more  correct  and  philosophical  view  of  this  subject. 

It  is  a  peculiarity  of  these  numbers  that  they  have  both  an 
abstract  and  a  concrete  signification.  The  denominate  numbers 
refer  to  time,  weight,  value,  etc :  these  things  are  not  tangible, 
material  things.  Time  is  nothing  that  you  can  touch  or  see, 
and  the  same  is  true  of  value  and  weight.  Length,  surface,  and 
volume  are  the  abstract  quantities  of  geometry.  Concrete 
things  possess  value  and  weight;  but  the  value  and  weight  are 
no  more  concrete  than  are  the  length,  surface,  or  volume. 
Denominate  numbers  are  therefore  not  concrete,  in  the  sense 
that  the  units  are  material  things.  They  are  concrete,  how- 
ever, in  the  sense  that  the  number  is  associated  with  some  par- 


NATURE   OF   DENOMINATE   NUMBERS.  493 

ticular  unit.  The  word  concrete,  from  con,  with,  and  cresco, 
I  grow,  means  literally  growing  or  united  together.  A  num- 
ber is  properly  concrete  when  it  is  associated  with  something 
numbered,  even  though  the  thing  itself  may  be  abstract  in  its 
nature.  It  is  not  the  character  of  that  which  is  numbered,  but 
the  fact  of  the  association  of  a  number  with  it,  that  makes  the 
number  concrete.  Thus,  the  number  four  used  independently 
of  any  object  is  abstract,  but  if  associated  with  some  object,  as 
boy,  yard,  pound,  etc.,  we  have  the  concrete  numbers  4  boys, 
4  yards,  etc. 

The  Standard  Units. — The  Units  of  Measure  are  of  as  many 
distinct  classes  as  there  are  distinct  kinds  of  continuous  quantity 
to  be  measured.  There  are  logically  four  different  kinds  of 
such  quantity,  and  consequently  there  are  four  distinct  classes 
of  units  of  measure.  These  units,  originating  by  chance,  were 
indefinite  and  variable,  and,  in  time,  were  found  unsuited  to  the 
purposes  of  a  civilized  people.  Science  then  took  the  matter  in 
hand  and  began  to  establish  standard  units,  having  definite 
values  derived  from  some  invariable  element  of  nature  by 
which  we  would  be  able  to  reconstruct  the  measures  at  any  time 
if  destroyed.  These  standard  units  were  so  related  to  each 
other  that,  having  fixed  one  of  them,  all  the  others  could  be 
derived  from  it.  The  fundamental  unit  agreed  upon  was  that 
of  length.  To  obtain  a  standard  unit  of  length  was  now  the 
question.  The  French  endeavored  to  fix  it  by  ascertaining  the 
distance  from  the  equator  to  the  poles,  and  taking  a  definite 
part  of  this  distance. 

This  was  done  by  Delambre  and  M^chain,  who  measured  an 
arc  of  the  meridian  between  Dunkirk  and  Barcelona,  and  gave 
the  standard  unit  of  length,  called  the  Meter.  The  English 
fixed  their  standard  unit  of  length  by  finding  the  length  of  a 
pendulum  which  vibrates  seconds.  The  latter  method  is 
regarded  as  the  most  convenient  in  practice,  though  it  is 
criticised  by  the  French  as  being  dependent  on  two  elements 
foreign  to  length, — that  is,  gravity  and  time.     From  the  unit 


494  THE   PHILOSOPHY    OF    ARITHMETIC. 

of  length,  however  obtained,  all  the  other  units,  except  that  of 
time,  may  be  derived.  It  will  be  noticed  that,  in  the  English 
method,  time  is  made  the  basis  of  the  system. 

The  Standard  Unit  of  Time  is  the  Day.  This  is  determined 
by  the  revolution  of  the  earth  upon  its  axis. 

The  Standard  Unit  of  Value,  in  the  United  States,  is  the 
Dollar.  It  is  determined  by  the  weight  of  the  metal  used  for 
money.  In  English  Money  the  standard  is  the  Pound,  deter- 
mined in  a  similar  manner. 

The  Standard  Unit  of  Weight  is  the  Troy  Pound.  It  is 
determined  by  taking  a  certain  number  of  cubic  inches  of 
distilled  water  at  a  given  temperature,  the  barometer  being  at 
a  certain  altitude.  The  Avoirdupois  pound  is  derived  from  the 
Troy  pound,  by  taking  7,000  Troy  grains.  The  Apothecaries' 
pound  is  the  same  as  the  Troy  pound. 

The  Standard  Unit  of  Length  is  the  Yard.  It  is  determined 
by  the  length  of  a  pendulum  which  vibrates  seconds  in  a  vacuum 
at  the  level  of  the  sea,  in  the  latitude  of  London.  Such  a  pen- 
dulum is  divided  into  391,393  equal  parts  and  360,000  of  these 
parts  taken  for  the  yard. 

The  Standard  Unit  of  Surface  is  the  Square  Yard  for 
ordinary  measurement,  and  the  Acre  for  land.  The  standard 
unit  of  Volume  is  the  Cubic  Yard  for  ordinary  measurement, 
and  the  Cord  for  wood.  These  are  derived  from  the  unit  of 
length. 

The  Standard  Unit  of  Capacity  is  the  Gallon  for  fluids,  and 
the  Bushel  for  dry  substances.  These  are  also  determined 
from  the  unit  of  length,  each  measure  consisting  of  a  certain 
number  of  cubic  inches. 

The  Standard  Unit  of  Angular  Measure  is  the  Right  Anglet 
or,  in  practice,  one  degree  of  a  circle. 

These  standard  units,  as  stated  above,  are  so  related  to  each 
other  that,  having  determined  one,  all  the  others  may  be  de- 
rived from  it.  Time  is  the  basis  of  the  English  system.  We 
first  find  the  Unit  of  Time  from  the  revolution  of  the  heavenly 


NATURE   OF   DENOMINATE   NUMBERS.  495 

bodies,  and  dividing  it  sufficiently  far  we  obtain  the  second. 
Having  the  second,  we  can  obtain  the  unit  of  length  by  ascer- 
taining the  length  of  a  seconds  pendulum  and  taking  a  definite 
part  of  it.  Having  the  unit  of  length,  we  readily  obtain  the  units 
of  area,  volume,  and  capacity.  The  standard  unit  of  weight  is 
obtained  by  taking  a  number  of  cubic  inches  of  water.  The 
unit  of  value  is  a  weight  of  gold  or  silver,  and  can  thus  be 
traced  back  to  its  origin  in  Time. 

Scales. — These  standard  units  are  divided  into  smaller  units, 
each  receiving  a  name  and  being  used  as  a  unit  of  measure,  and 
these  are  again  subdivided  in  a  similar  manner.  Multiples  of 
the  standard  units  are  also  used  as  new  measures,  multiples  of 
these  in  the  same  way,  the  series  being  continued  as  far  as 
convenient.  This  gives  a  series  of  measures  for  the  estimation 
of  the  same  kind  of  quantity,  forming  a  scale  of  numbers. 
Any  number  expressed  in  two  or  more  terms  of  such  a  scale, 
constitutes  what  is  called  a  Compound  Number.  A  Compound 
Number,  therefore,  consists  of  several  denominate  numbers  of 
the  same  kind  of  quantity. 

Since  the  different  standards  of  comparison,  their  multiplica- 
tion and  division,  originated  at  different  times  and  under  dif- 
ferent circumstances,  it  is  natural  that  these  scales  should  be 
irregular  and  without  system.  Sometimes  the  scale  of  increase 
is  by  twos,  fours,  etc.;  sometimes  by  fours,  twelves,  etc.;  and 
again  by  12,  3,  5^,  etc.  This  irregularity  will  be  most  clearly 
seen  by  comparing  any  scale  of  compound  numbers  with  the 
decimal  scale.  Take,  for  instance,  the  scale  of  English  Money 
and  write  it  beside  the  decimal  scale ;  thus, 

T.  h.  t.  u.  £  s.  d.  qr. 

1111  1111. 

In  the  decimal  scale  the  second  unit  is  ten  times  the  first,  the 
third  is  ten  times  the  second,  etc.  In  the  scale  of  English 
Money,  the  second  unit  is  four  times  the  first,  the  third  is 
twelve  times  the  second,  the  fourth  is  twenty  times  the  third. 
The  same  irregularity  obtains  among  all  the  other  scales  of 
denominate  numbers. 


496  THE    PHILOSOPHY    OF    ARITHMETIC. 

This  irregularity  of  scale  is  a  serious  defect  of  our  measures 
of  value,  weight,  etc.,  viewed  from  either  a  scientific  or  a  practi- 
cal standpoint.  Science  dictates  that  the  multiples  and  divi- 
sions of  the  standard  units  should  be  uniform  and  correspond 
to  the  scale  of  notation,  which  with  us  would  make  the  scales 
decimal.  This  would  enable  us  to  add,  subtract,  multiply,  and 
divide  compound  numbers  just  as  we  do  abstract  numbers. 
Prance  has  adopted  this  method  in  all  its  measures,  and  the 
United  States  in  its  currency.  This  method  has  many  very 
important  advantages;  the  only  objection  to  it  arises  from  the 
decimal  base  of  our  system  of  notation,  since  the  simple  frac- 
tional parts,  thirds,  fourths,  and  sixths,  are  not  aliquot  parts 
of  ten.  With  a  system  of  notation  whose  base  was  twelve,  the 
method  would  be  much  more  convenient  in  practice. 


CHAPTER  II. 

MEASURES   OF   EXTENSION. 

TIIE  Measures  of  Extension  are  of  three  kinds,  measures  of 
Length,  of  Surface,  and  of  Volume  or  Capacity.  This 
division  arises  from  the  fact  that  extension  possesses  but  three 
elements;  length,  breadth,  and  thickness.  To  these,  however, 
must  be  added  Angular  or  Circular  Measure,  which  is  the  degree 
of  divergence  of  two  lines,  or  the  length  of  an  arc  of  a  circum- 
ference used  to  indicate  this  divergence. 

The  standard  units  of  this  measure  were  originally  derived 
from  the  natural  objects  of  the  material  world.  In  their  origin 
they  followed  the  general  law  of  mental  and  scientific  develop- 
ment  from  the  concrete  to  the  abstract.     Thus  originated  the 

foot,  the  cubit,  the  span,  the  fathom,  the  barleycorn,  the  hair- 
breadth, and  other  measures  taken  from  parts  of  the  human 
body,  or  from  natural  objects  which  possess  a  certain  mean 
value  sufficiently  definite  to  answer  the  purposes  of  a  rude  so- 
ciety. The  same  is  true  for  all  the  other  measures  in  denomi- 
nate numbers.  Among  the  measures  of  weight  we  have  the 
grain,  originally  a  grain  of  wheat;  the  pennyweight,  which 
was  the  weight  of  the  English  penny,  etc.  For  measures  of 
value  some  nations  use  cattle ;  others,  pigs ;  the  Icelanders, 
dried  fish ;  the  American  Indians,  the  skins  of  animals.  The 
first  measures  of  time  were  derived  from  the  revolutions  of  the 
heavenly  bodies :  thus  month,  derived  from  moon-eth,  was  orig- 
inally the  time  measured  by  the  revolution  of  the  moon.  The 
subject  being  interesting  and  instructive,  will  be  treated  some- 
what in  detail  The  principal  authorities  followed  are  Spencer 
and  the  popular  encyclopedias. 

32  ( 497  ) 


498  THE   PHILOSOPHY    OF   ARITHMETIC. 

Nearly  all  of  the  original  measures  of  length  were  taken  from 
the  parts  of  the  human  body.  The  Hebrew  cubit  was  the 
length  of  the  forearm  from  the  elbow  to  the  end  of  the  middle 
finger ;  the  smaller  Scriptural  dimensions  are  expressed  in  hand- 
breadths  and  spans.  The  Egyptian  cubit,  which  was  similarly 
derived,  was  divided  into  digits  which  were  finger-breadths  ; 
and  each  finger-breadth  was  regarded  as  equal  to  four  grains  of 
barley  placed  breadth-wise.  Other  ancient  measures  were  the 
orgyia,  or  stretch  of  the  arms,  the  pace,  the  palm,  etc.  So 
general  and  persistent  was  the  use  of  these  natural  units  in 
the  East,  that  even  now  some  of  the  Arab  tribes  mete  out  cloth 
by  the  forearm,  with  the  addition  of  the  breadth  of  the  other 
hand,  which  marks  the  end  of  the  measure.  The  width  of  the 
thumb  was  in  like  manner  added  at  the  end  of  the  yard  by 
English  clothiers;  and  it  is  not  unusual  to  see  women,  even 
in  this  country,  test  the  number  of  yards  in  a  purchase  by 
measuring  it  from  the  chin  to  the  end  of  the  fingers.  The  foot 
was  used  by  the  Romans,  and  is  still  a  measure  in  Europe  and 
America,  its  length  in  different  places  varying  not  much  more 
than  men's  feet  vary.  The  height  of  horses  is  expressed  in 
hands,  and  the  depth  of  water  in  fathoms,  the  length  of  the 
two  arms  extended.  The  inch  is  supposed  to  be  the  length  of 
the  terminal  joint  of  the  thumb,  as  appears  in  the  French  lan- 
guage, where  pouce  means  both  thumb  and  inch.  We  have 
also  the  inch  divided  into  barleycorns,  the  inch  being  regarded 
as  equal  to  three  of  these. 

So  completely  indeed  have  these  natural  dimensions  served 
as  the  basis  of  all  mensuration,  that  it  is  only  by  means  of  them 
that  we  can  form  any  estimate  of  some  of  the  ancient  distances. 
For  example,  the  length  of  a  degree  on  the  earth's  surface,  as 
determined  by  the  Arabian  astronomers,  shortly  after  the  death 
of  Haroun-al-Raschid,  was  fifty-six  of  their  miles.  We  know 
nothing  of  their  mile,  further  than  that  it  was  4,000  cubits; 
and  whether  these  were  sacred  cubits  or  common  cubits,  would 
remain  doubtful,  but  that  the  length  of  the  cubit  is  given  as 


MEASURES   OF   EXTENSION.  499 

twenty-seven  inches,  and  each  inch  defined  as  the  thickness  of 
six  barley-grains.  Thus  one  of  the  earliest  measurements  of  a 
degree  comes  down  to  us  in  barley-grains.  Not  only  did 
oiganic  lengths  furnish  those  approximate  measures  which 
satisfied  men's  needs  in  ruder  ages,  but  they  furnished  also  the 
standard  measures  required  in  later  times.  One  instance 
occurs  in  the  reign  of  Henry  I.,  who,  to  remedy  the  irregulari- 
ties then  prevailing,  commanded  that  the  ulna,  or  ancient  ell, 
which  answers  to  the  modern  yard,  should  be  made  the  exact 
length  of  his  own  arm. 

As  civilization  advanced,  the  inaccuracy  of  such  variable 
measures  as  the  foot,  cubit,  etc.,  were  felt,  and  the  necessity  of 
adopting  more  precise  standards  became  apparent.  To  do  this, 
it  was  necessary  to  find,  among  the  objects  of  nature,  a  standard 
perfectly  definite,  and  at  the  same  time  invariable  and  accessible 
to  all  mankind.  To  obtain  such  an  object  was  a  matter  of 
no  inconsiderable  difficulty.  In  fact,  nature  presents  only  two 
or  three  elements  which,  with  the  aid  of  a  profound  science 
and  a  refined  knowledge  of  the  arts,  can  be  made  subservient 
to  the  purpose,  and  none  at  all  without  such  aid.  The  earth 
is  nearly  a  solid  of  revolution,  and  its  form  and  absolute  mag- 
nitude are  presumed  to  remain  the  same  in  all  ages ;  hence  the 
distance  from  the  equator  to  the  poles  is  an  invariable  quantity, 
and  some  definite  part  of  this  distance,  if  exactly  ascertained, 
might  be  taken  as  a  standard  unit  of  length.  The  force  of 
gravity  at  the  earth's  surface  is  constant  at  any  given  place, 
and  is  nearly  the  same  at  all  places  under  the  same  parallel 
of  latitude,  and  at  the  same  height  above  the  level  of  the 
sea;  hence  the  length  of  a  pendulum  which  makes  a  given 
number  of  vibrations  in  a  given  time  is  constant,  and  might  be 
used  to  determine  a  standard  unit  of  length. 

These  two  elements,  the  length  of  a  degree  of  the  meridian, 
and  the  length  of  a  seconds  pendulum,  are  the  only  ones  fur- 
nished by  nature  which  have  yet  been  used  as  a  basis  of  a  sys- 
tem of  measures.     One  or  two  others  have  been  suggested,  as 


500  THE    PHILOSOPHY   OP   ARITHMETIC. 

the  height  through  which  a  body  falls  in  a  second  of  time,  and 
the  perpendicular  height  through  which  a  barometer  must  be 
carried  till  the  mercurial  column  sinks  a  determinate  part,  for 
example,  one-thirtieth  of  its  own  length ;  but  these  distances  are 
not  so  susceptible  of  being  accurately  determined  as  the  terres- 
trial degree,  or  the  length  of  the  seconds  pendulum.  Mouton, 
an  astronomer  of  Lyons,  about  1670,  proposed  as  a  standard,  a 
geometrical  foot,  of  which  a  degree  of  the  earth's  circumference 
should  contain  600,000 ;  and  remarked  that  a  pendulum  of  this 
Jength  would  make  3959  J-  vibrations  in  a  half  hour.  In  1671, 
Picard  proposed  a  similar  method ;  and  Huygens  first  suggested 
the  pendulum  as  the  unit  or  standard  of  measures.  No  attempt 
was  made  to  establish  a  regular  system  of  measures  until  the 
time  of  the  French  Revolution,  when  a  system  of  weights  and 
measures,  referred  to  the  terrestrial  degree,  and  accommodated 
to  our  arithmetical  scale,  was  adopted  in  that  country. 

English  Standard  Measures. — The  English  standard  unit 
of  length  is  the  yard,  which,  in  the  reign  of  Henry  I.,  was  de- 
termined by  the  length  of  the  king's  arm.  An  act  of  Edward 
II.,  1324,  provides  that  the  length  of  3  barleycorns,  round  and 
dry,  shall  make  an  inch,  12  inches  a  foot,  etc.  The  difficulty 
of  determining  how  much  of  the  end  of  the  grain  should  be 
removed  to  render  it  "round"  makes  this  standard  rather 
indefinite.  No  record  exists,  however,  of  the  actual  construc- 
tion of  standard  units  based  upon  the  use  of  barleycorns.  In 
time,  for  the  purpose  of  securing  some  uniformity  among  the 
ordinary  measures,  certain  standards  were  placed  in  the  Ex- 
chequer, with  which  all  rods  were  required  to  be  compared  be- 
fore they  were  stamped  as  legal  measures.  The  oldest  of  the 
standards  in  existence  dates  from  the  reign  of  Henry  VII.,  but 
it  has  long  been  disused.  There  was  another  similar  rod  of 
the  same  date,  called  an  ell,  though  it  was  not  established  as  a 
legal  measure,  but  was  conventionally  regarded  as  equal  to  a  yard 
and  a  quarter.  That  which,  till  the  year  1824,  was  considered 
as  the  legal  standard,  was  a  brass  rod  of  the   breadth   and 


MEASURES   OF   EXTENSION.  501 

thickness  of  about  half  an  inch,  placed  there  in  the  time  of 
Elizabeth.  This  standard  was  illy  fitted  for  the  purpose  de- 
signed. "A  common  kitchen  poker,  filed  at  the  end  in  the 
rudest  manner,  by  the  most  bungling  workman,  would  make  as 
good  a  standard.  It  has  been  broken  asunder  aud  the  two 
pieces  dovetailed,  but  so  badly  that  the  joint  is  nearly  as  loose 
as  that  of  a  pair  of  tongs." 

In  the  year  1742,  some  Fellows  of  the  Royal  Society,  and 
members  of  the  Academy  of  Sciences  at  Paris,  proposed  to 
have  accurate  standards  of  the  ''measures  and  weights"  of  both 
nations  made  and  carefully  examined,  in  order  that  the  results 
of  the  scientific  experiments  in  England  and  France  might  be 
correctly  compared.     The  committee  who  undertook  the  mat- 
ter, besides  the  standard  in  the  Exchequer,  found  some  others 
which  were  considered  of  good,  if  not  of  equal  authority.     At 
Guildhall  they  found  two  standards  of  length,  which  were  only 
two  beds  or  matrices,  one  of  a  yard,  the  other  of  an  ell,  cut  out 
of  the   edges   of  a   brass   bar,    like   that   of  the    Exchequer. 
Another,  kept  in  the  Tower  of  London,  was  a  solid  brass  rod, 
about  seven-tenths  of  an  inch  square,  and  41  inches  long,  on 
■one  side  of  which  was  a  yard  divided  into  inches.     Another, 
belonging  to  a  clockmakers'  company,  derived  from  the   Ex- 
chequer in  167 1,  was  a  brass  rod  of  eight  sides,  on  which  the 
length  of  the  yard  was  expressed  by  the  distance  between  two 
pins  or  small  checks.     The  committee  selected  the  standard  in 
the  Tower,  and  Mr.  George  Graham,  a  celebrated  clockmaker, 
in  1742,  laid  off  from  it,  with  great  care,  the  length  of  the  yard 
on  two' brass  rods  which  were  then  sent  to  the  Academy  of 
Scieuces  at  Paris ;    on  these,  in  like  manner,  was  set  off  the 
measure  of  the  Paris  half  toise.     One  of  these  was  kept  at 
Paris,  the  other  was  returned  to  the  Royal  Society,  where  it 
still  remains;    but   unfortunately  it  was  not  stated    at  what 
temperature  the  toise  was  set  off,  so  that  the  comparison  is  now 
of  little  value. 

In  1758,  a  committee  of  the  House  of  Commons  recommended 


502  THE   PHILOSOPHY    OF    ARITHMETIC. 

that  a  rod  which  had  been  made  at  their  order,  by  Mr.  Bird„ 
from  that  of  the  Royal  Society,  and  marked  "  Standard  Yard 
of  1758,"  should  be  declared  the  legal  standard  of  all  measures 
of  length.  In  the  following  year  another  committee  was  formed 
on  the  subject,  which  concurred  in  the  Bird  standard,  and  au- 
thorized Mr.  Bird  to  make  a  copy  of  the  former  rod,  which  he 
completed  in  1760.  No  further  action  was  taken  by  the  govern- 
ment until  1824,  when  a  very  thorough  revision  was  attempted. 

Stimulated  by  the  scientific  efforts  of  the  French  philoso- 
phers, the  English  turned  their  attention  to  the  establishment 
of  an  invariable  standard  unit  of  length,  and  selected  as  a  basis 
the  seconds  pendulum  at  London.  The  length  of  such  a  pen- 
dulum had  been  determined  as  early  as  1742,  by  George  Gra- 
ham, to  be  39.13  inches,  and  used  for  the  construction  of  a  stand- 
ard yard.  Reports  were  made  in  1816,  1818,  and  1820,  to  the 
House  of  Commons,  based  on  experiments  and  comparisons,  in 
which  Wollaston,  Dr.  Young,  Capt.  Kater,  and  Prof.  Playfair, 
took  a  prominent  part.  This  led  to  the  adoption  of  the  imperial 
measures  and  standards  under  George  IV.,  which  took  effect 
January  1,  1826 :  these  standards  were  retained  by  the  enact- 
ments under  William  IV.,  which  took  effect  January  1,  1836.  In 
the  imperial  measures,  the  yard  copied  from  the  standard  of  1760 
was  to  be  of  brass,  and  measured  at  the  temperature  of  62°  F., 
while  its  length  was  further  defined  by  declaring  that  the  pen- 
dulum beating  seconds  of  mean  time  in  the  latitude  of  London 
at  the  temperature  named,  in  a  vacuum  at  the  level  of  the  sea, 
should  be  39.1393  inches  of  the  above  standard.  From  this 
standard  measure  of  length,  all  the  other  measures,  of  weighty 
capacity,  etc.,  were  also  established. 

Soon  after  the  standards  were  prepared,  they  were  destroyed 
by  the  burning  of  the  Houses  of  Parliament,  1834;  but  fortu- 
nately the  Astronomical  Society  had  procured  a  most  carefully 
prepared  copy  of  the  imperial  standard  yard,  and  the  mint  was 
in  possession  of  an  exact  copy  of  the  pound,  so  that  it  was 
possible  to  reproduce  the  lost  standards  with  great  precision. 


MEASURES  OF   EXTENSION.  503 

In  1838  a  commission  was  appointed,  of  which  Airy,   Baily, 
Herschel,  Lubbock,  and  Shepherd  were  members,  which,  after 
a  very  thorough  investigation  of  the  matter,  reported  in  1841, 
that,  since  the  passage  of  the  act  of  George  IV.,  several  ele- 
ments of  reductiou  of  the  pendulum  experiments,  on  which 
some  of  its  provisions  were  based,  had  been  found  to  be  doubt- 
ful or  erroneous,  there  having  been  defects  in  the  agate  planes 
of  the  pendulum  used  by  Capt.  Kater,  and  errors  in  finding  its 
specific  gravity,  and  in  reductions  for  buoyancy  of  the  air  and 
for   elevation   above   the  level  of  the  sea.     They  concluded 
that  the  course  prescribed  in  the  act  would  not  produce  the 
original  yard ;  that  the  condition  that  the  yard  was  to  be  a 
certain  brass  rod  was  the  best  that  could  be  adopted  ;  aud  that, 
by  the  aid  of  the  Astronomical  Society's  scale,  and  a  few  other 
highly  accurate  copies  known,  the  standard  could  be  restored 
without  sensible  error.      Mr.  Baily  was  selected   to   prepare 
the  new  standard,  with  five  copies  of  the  preceding  on  which 
to  base  his  comparison;  and  on  his  death  iu  1844,  Mr.  Sheep- 
shanks continued  the  necessary  observations,  executing  himself; 
in  the  course  of  this  labor,  about  200,000  micrometric  measure- 
ments.    He  prepared   several  standard  copies,  each  being  a 
square  inch  bar,  of  a  bronze  consisting  of  copper  with  a  small 
percentage  of  tin  and  zinc,  38  inches  in  length,  with  half-inch 
wells  sunk  to  the  middle  of  the  bar,  one  inch  from  each  end,  in 
which  the  lines  defining  the  yard  are  drawn  on  gold  plugs. 
Six  of  these  were  finally  selected  and  reported  by  the  commis- 
sioners in  March,  1854 ;  and  one  of  these,  marked  "  Bronze  19,n 
was  selected  as  the  parliamentary  standard  yard,  the  remaining 
five   being  deposited,  along  with  copies  of  the  standard  of 
weight,  with  as  many  public  institutions  and  scientific  bodies. 
These  standards  were  legalized  in  July,  1855;  and  provisions 
made   that   in   case   of  loss   of  the   parliamentary   copy,   the 
standard    should    be  restored   by   comparison   of    the    other 
selected   copies,  or   such  as  might  be  available.     From  this 
statement  it  is  seen  that  the  latest  verdict  of  science  is  adverse 


504  THE    PHILOSOPHY    OF    ARITHMETIC. 

to  the  practicability  of  basing  a  system  of  measures  on  any  in- 
variable natural  unit  of  length. 

The  measures  of  the  American  colonies  were  the  same  as 
those  of  the  mother  country  at  the  corresponding  period. 
Variations  naturally  grew  up  in  the  different  colonies,  and  the 
several  measures  in  use  being  adopted  with  little  or  no  change 
when  they  became  states,  the  discrepancies  continued  to  exist. 
On  the  3d  of  March,  1817,  John  Quincy  Adams  was  commis- 
sioned by  a  resolution  of  the  Senate,  to  examine  the  subject  of 
the  weights  and  measures  of  the  United  States,  and  also  to 
consider  the  desirableness  of  adopting  the  French  system,  or 
one  similar  to  it.  The  standards  employed  in  the  various 
custom-houses  were  examined  and  carefully  measured  during 
1819  and  1820,  under  his  direction,  and  in  a  report  published 
in  1821  he  showed  that  very  considerable  discrepancies  existed 
between  the  measures  of  the  several  states,  and  often  within 
the  same  state.  After  a  careful  review  of  the  French  system, 
he  reported  unfavorably  to  its  adoption,  on  account  of  the  dif- 
ficulty of  the  change  and  the  essential  inconvenience  of  a  deci- 
mal system. 

On  the  29th  of  May,  1830,  the  Senate  directed  a  new  com- 
parison of  the  weights  and  measures  in  use  at  the  different 
custom-houses.  This  examination  was  made  by  Prof.  Hassler, 
who  found  that,  though  considerable  discrepancy  existed,  the 
mean  value  corresponded  closely  with  the  English  standards 
verified  in  1776.  Under  his  supervision  accurate  copies  of  the 
received  standards  of  weights  and  measures  were  supplied  to 
all  the  custom-houses;  and  by  act  of  Congress,  June  14,  1836, 
the  Secretary  of  State  was  directed  to  have  sent  to  the  gover- 
nors of  all  the  states  a  complete  set  of  all  standards,  for  the 
use  of  the  several  states.  These,  as  well  as  accurate  balances 
for  adjusting  the  weights,  were  supplied  ;  and  the  statutory 
standards  of  every  state  have  been  made  to  conform  to  the 
standards  so  furnished.  No  enactments  were  made  concerning 
the  old  English  standards  of  length,  which  have  come  down  to 


MEASURES   OF   EXTENSION.  505 

us,  as  they  were  necessarily  in  force  unless  changed  by  legis- 
lative enactment. 

It  is  to  be  noticed  that  the  American  yard  was  taken  from 
a  scale  made  for  the  United  States  by  Troughton,  which  was 
supposed  to  be  identical  with  the  old  standard,  and  with  the 
Astronomical  Society's  scale,  but  which  had  never  been  directly 
compared  with  either.  When  this  comparison  was  made  with 
the  bronze  bar  No.  11,  which  had  been  presented  to  the  United 
States  by  the  British  Government,  the  yard  on  the  Troughton 
scale  was  found  to  be  nearly  ToVo"  °f  an  mcn  t0°  l°n&  5  and 
hence  all  the  copies  furnished  to  the  states  are  subject  to  that 
minute  correction,  since  the  British  yard  is  unquestionably  the 
only  authentic  representative  of  the  old  standard  from  which 
our  measures  are  derived.  In  1866  Congress  authorized  the 
use  of  the  metric  system,  but  the  present  indications  are  that 
it  will  be  very  tardily  introduced. 

Measures  of  extension,  as  already  stated,  are  of  three  dis* 
tinct  classes:  measures  of  Length,  measures  of  Surface,  and 
measures  of  Volume  or  Capacity.  Measures  of  Length  are 
of  several  different  kinds ;  the  common  Long  Measure,  Sur- 
veyors' Measure,  etc.  Measures  of  Surface  include  the  ordinary 
Surface  Measure,  aud  Surveyors'  Square  Measure.  Measures 
of  Volume  include  the  ordinary  Cubic  Measure,  and  Meas- 
ures of  Capacity.  Measures  of  Capacity  embrace  those  of 
Liquids  and  Dry  Substances.  A  few  facts  in  addition  to  those 
already  given  will  be  stated. 

Long  Measure. — Long  Measure  is  the  measure  of  length, 
applied  to  the  measurement  of  the  length,  breadth,  or  thickness 
of  objects ;  also  to  heights  and  distances.  The  unit,  as  already 
explained,  is  the  yard,  which  is  identical  with  the  Imperial 
yard  of  Great  Britain,  divisions  and  multiples  of  which  give 
an  irregular  scale  of  derived  units.  These  units  were  nearly 
all  originally  derived  from  parts  of  the  human  body.  Thus, 
foot  is  from  the  human  foot;  yard  was  a  rod  or  shoot;  rod  is 
from  a  measuring  stick  or  rod ;  furlong  is  from  fur,  furrow, 
22 


506  THE    PHILOSOPHY    OF    ARITHMETIC. 

and  long,  meaning  long,  or  the  length  of  a  furrow;  mile  is  from 
mille  passuum,  a  thousand  paces  ;  span  is  the  space  measured 
by  the  thumb  and  little  finger  extended;  cubit,  the  forearm ; 
fathom,  the  length  of  the  two  arms  extended.  The  inch  is 
supposed  by  some  to  be  derived  from  tne  terminal  joint  of  the 
thumb,  though  late  etymologists  get  it  from  uncia,  the  twelfth 
part.  The  inch  was  formerly  divided  into  three  equal  parts, 
called  barleycorns,  the  length  of  a  grain  or  kernel  of  barley. 

The  geographic  mile  is  equal  to  1  minute  of  one  of  the  great 
circles  of  the  earth,  hence  it  equals  ^  of  -^hs  of  the  circumfer- 
ence of  the  earth,  which  equals  about  1.15  statute  miles.  The 
knot,  used  in  measuring  distances  at  sea,  is  equivalent  to  a  geo- 
graphic mile.  The  English  mile  is  the  same  as  that  of  the 
United  States.  The  German  short  mile  equals  6857  yards,  or 
about  3 t9q  statute  miles ;  the  German  long  mile  equals  10125 
yards,  or  about  5|  statute  miles  ;  the  Prussian  mile  equals  823T 
yards,  or  about  4^  statute  miles. 

A  degree  of  longitude  at  any  point  is  3^  of  the  circle  pass- 
ing through  the  latitude  of  that  point ;  and  as  these  circles 
diminish  as  we  pass  from  the  equator,  the  degrees  of  longitude 
will  diminish.  Thus,  at  the  equator,  the  length  of  a  degree  of 
longitude  is  about  69g-  statute  miles ;  at  25°  of  latitude,  62^ 
miles ;  at  40°  of  latitude,  53  miles  ;  at  42°,  51 J  miles ;  at  49°,  45£ 
miles;  at  60°,  34^  miles,  etc.  A  degree  of  latitude  also  varies, 
being  68.72  miles  at  the  equator ;  from  68.9  to  69.05  miles  in 
middle  latitude  ;  and  from  69.30  to  69.34  miles  in  the  polar 
regions. 

Surveyors'  Linear  Measure  is  used  by  surveyors  and  engi- 
neers in  measuring  the  dimensions  of  land,  distances,  etc.  The 
unit  is  the  chain,  called  Gunter's  chain,  from  Edmund  Gunter, 
the  reputed  inventor,  an  English  mathematician,  born  1581,  died 
1626.  It  is  4  rods,  or  66  feet  in  length,  and  is  divided  ac- 
cording to  the  decimal  system  into  100  equal  parts  called  links. 
This  division  reduces  all  the  calculations  to  the  decimal  system> 
and  thus  greatly  simplifies  the  operations. 


MEASURES  OF   EXTENSION.  507 

The  denomination  rods  is  seldom  used  by  surveyors,  dis- 
tances being  represented  in  chains  and  links.  Since  each  link 
is  1_iT  of  a  chain,  the  number  of  links  is  generally  expressed  as 
a  decimal;  thus  5  chains  and  47  links  are  written  5.47  chains. 
Engineers  generally  use  a  chain  100  feet  long,  containing  120 
links,  each  10  inches  in  length.  Mariners'  Measure  is  used  by 
seamen  in  measuring  distances,  the  depth  of  the  sea,  etc.  The 
old  method  of  Cloth  Measure  is  practically  obsolete,  the  divi 
sion  of  the  yard  being  into  haloes,  quarters,  etc.,  and  cloth 
being  thus  sold  instead  of  by  the  nail  or  inch.  At  the  custom- 
houses, the  yard  is  divided  into  tenths,  hundredths,  etc. 

Surface  Measure. — Surface  or  Square  Measure  is  used  in 
measuring  surfaces,  as  land,  boards,  amount  of  painting,  paper- 
ing, plastering,  paving,  etc.  The  term  Perch  is  from  the 
French  perche,  a  pole;  rood  is  supposed  to  be  a  corruption 
of  rod;  acre  was,  primarily,  an  open  plowed  or  sowed  field. 
The  unit  for  land  is  the  acre ;  for  other  surfaces  it  is  usually 
the  square  yard.  The  Perch  is  a  surface  equivalent  to  a 
square  rod.  The  Rood  is  less  used  than  formerly.  A  square 
piece  of  land,  measuring  209  feet,  or  about  70  paces,  on  each 
side,  equals  very  nearly  1  acre. 

Surveyors'  Square  Measure  is  used  by  surveyors  in  comput 
ing  the  area  or  contents  of  land.  The  perch  and  rood  are  not 
so  much  used  as  formerly,  the  contents  of  land  being  commonly 
estimated  in  square  miles,  acres,  and  hundredths.  Govern 
ment  lands  are  divided  by  parallels  and  meridians  into  town- 
ships, which  contain  36  square  miles  or  sections,  and  each  sec- 
tion is  subdivided  into  quarter-sections.  Hence  640  acres 
make  a  section,  and  160  acres  a  quarter-section.  A  hide  of 
land,  which  is  spoken  of  by  ancient  writers,  is  100  acres. 

Cubic  Measure. — Cubic  or  Solid  Measure  is  used  in  measur- 
ing things  which  have  length,  breadth,  and  thickness.  The 
fundamental  unit  seems  to  be  the  cubic  foot.  The  other  units 
are  not  in  a  regular  scale,  but  are  parts  or  multiples  of  the 
cubic   foot.     The    old    unit   for   measuring  timber    was   the 


508  THE    PHILOSOPHY    OF    ARITHMETIC. 

ton.  Round  timber,  when  squared  for  use,  is  supposed 
to  lose  \ ;  hence  a  ton  of  round  timber  is  said  to  contain 
such  a  quantity  of  timber  in  its  rough  or  natural  state  as, 
when  hewn,  will  make  40  cubic  feet,  and  is  supposed  to  be 
equal  in  weight  to  50  cubic  feet  of  hewn  timber.  Timber  is 
now  generally  sold  by  board  measure,  the  ton  beiug  nearly  ob- 
solete. A  cord  of  wood  is  a  pile  8  feet  long,  4  feet  wide,  and 
4  feet  high.  A  cord  foot  is  a  part  of  this  pile  1  foot  long  ;  it 
equals  16  cubic  feet. 

A  perch  of  stone  or  masonry  is  16 J  feet  long,  1J  feet  wide, 
and  1  foot  high;  it  contains  24 J  cubic  feet.  A  cubic  yard  of 
earth  is  called  a  load.  A  square  of  earth  is  a  cube  measuring 
6  feet  on  a  side,  and  contains  216  cubic  feet.  In  civil  engineer- 
ing, the  unit  is  the  cubic  yard,  to  which  all  estimates  for  ex- 
cavations and  embankments  are  reduced.  The  measurements 
are  taken  with  a  line  divided  into  feet  and  decimals  of  a  foot. 
A  Register  Ton  is  the  standard  for  estimating  the  capacity  or 
tonnage  of  vessels,  and  is  100  cubic  feet.  A  Shipping  Ton, 
used  in  estimating  cargoes,  in  the  United  States  is  40  cubic 
feet;  in  England,  42  cubic  feet. 

Liquid  Measures. — Liquid  Measures  are  used  for  measuring 
all  kinds  of  liquids.  They  are  of  three  classes:  Wine  Measure, 
Beer  Measure,  and  Apothecaries'  Fluid  Measure.  Wine  meas- 
ure is  now  used  for  measuring  nearly  all  kinds  of  liquids.  It 
was  called  wine  measure  because  it  was  used  to  measure  wine, 
instead  of  beer,  which  was  measured  by  another  measure. 

The  standard  unit  of  wine  measure  is  the  gallon.  It  is 
intended  to  represent  the  old  wine  gallon  of  231  cubic  inches, 
but  is  denned  as  containing  58,372.2  grains  of  distilled  water 
at  its  maximum  density,  weighed  in  air  of  the  temperature  of 
62°  F.,  and  barometric  pressure  of  30  inches.  This  is  identical 
with  the  old  Winchester  gallon  of  England,  so  called  from  the 
standard  having  been  formerly  kept  at  Winchester,  England. 
The  Imperial  gallon,  adopted  by  Great  Britain  in  1836,  is 
defined  as  containing  10  pounds  avoirdupois  of  distilled  water. 


MEASURES   OF   EXTENSION.  509 

at  a  temperature  of  62°  Fahrenheit,  the  barometer  standing  at 
30  inches.  It  contains  277.274  cubic  inches.  The  American 
gallon  is  thus  0.83311  of  the  Imperial  gallon,  or  about  6  of  the 
former  equal  5  of  the  latter. 

There  were  in  England,  from  1650  to  1688,  three  different 
measures  of  the  wine  gallon.  The  one  of  general  usage  con- 
tained 231  cubic  inches  to  the  gallon.  Another,  the  customary 
standard  at  Guildhall,  supposed  to  be  of  the  same  capacity, 
was  found  by  measurement  to  contain  only  224  cubic  inches. 
A  third,  the  real  and  legal  standard,  preserved  at  the  Treasury, 
contained  282  cubic  inches.  The  corn  gallon  differed  from  any 
of  these,  being  268.6  cubic  inches.  Some  suppose  the  gallons 
of  231  and  282  cubic  inches  to  have  originated  under  separate 
enactments,  the  latter  from  one  of  Henry  VII.,  directing  that 
the  gallon  contain  8  pounds  of  wheat;  but  Oughtred  holds 
that  the  larger  or  beer  gallon  was  allowed  for  liquids  which 
yield  froth,  as  beer,  etc.,  and  the  smaller  for  such  liquids,  as 
wine  and  oil,  which  do  not  froth,  and  thus  their  exact  volume 
is  immediately  indicated. 

The  term  gill  is  from  Low  Latin  gilla,  a  drinking  glass  ; 
pint  is  from  the  Anglo-Saxon  pyndan,  to  shut  in,  to  pen,  or 
from  the  Greek  pinto,  to  drink  ;  quart  is  from  the  Latin  quartus, 
a  founh.  The  derivation  of  gallon  is  not  clear;  in  the  French, 
a  galon  is  a  grocer's  box.  Barrels  and  hogsheads  are  of  varia- 
ble capacity.  The  values  given  in  the  tables  are  used  in 
estimating  the  capacity  of  wells,  cisterns,  vats,  etc.  In  Mas- 
sachusetts the  barrel  is  estimated  at  32  gallons.  A  pint  of 
water  weighs  nearly  one  pound,  hence  the  old  adage,  "A 
pint's  a  pound,  the  world  around." 

Ale  or  Beer  Measure  was  formerly  used  in  measuring  ale, 
beer,  and  milk.  It  was  named  Beer  Measure  from  its  being 
so  extensively  used  in  measuring  beer,  in  distinction  from  the 
measure  used  for  wine  and  oil,  etc.  The  measure  was  greater 
than  wine  measure,  as  beer  was  less  costly  than  wine,  or,  as 
some  have  supposed,  on  account  of  the  frothing  of  beer.     The 


510  THE   PHILOSOPHY    OF   ARITHMETIC. 

unit  is  the  gallon,  which  contains  282  cubic  inches,  or  10.179933 
pounds  avoirdupois  of  distilled  water.  This  measure  is  going 
out  of  use ;  milk  and  also  beer  and  ale  are  now  generally  meas- 
ured by  Wine  Measure. 

Apothecaries'  Fluid  Measure  is  used  for  measuring  liquids  in 
preparing  medical  prescriptions.  Minim  is  from  the  Latin 
minimus,  the  least,  the  minim  being  the  smallest  fluid  measure 
used.  Several  of  the  other  terms  are  formed  by  prefixing  fluid 
to  the  terms  of  Apothecaries'  Weight.  Gong,  is  the  abbre- 
viation of  congius,  the  Latin  for  gallon.  0.  is  the  initial  of 
octarius,  the  Latin  for  one-eighth,  the  pint  being  one-eighth  of  a 
gallon.  In  estimating  the  quantity  of  fluids,  45  drops  equal 
about  a  fluid  drachm ;  a  common  teaspoon  holds  about  1  fluid 
drachm  ;  a  common  tablespoon  about  ^  a  fluid  ounce ;  a  wine- 
glass about  1^  fluid  ounces;  a  common  teacup  about  4  fluid 
ounces. 

Dry  Measure. — Dry  Measure  is  used  for  measuring  dry  sub- 
stances, such  as  grain,  fruit,  salt,  coal,  etc.  The  unit  of  dry 
measure  is  the  bushel,  which  is  divided  into  pecks,  quarts,  etc. 
The  term  bushel  is  derived  from  a  word  meaning  box.  The 
term  peck  is  supposed  to  be  a  corruption  of  pack,  or  to  be  de- 
rived from  the  French  picotin,  a  peck.  The  Imperial  bushel 
of  England,  by  the  act  of  George  IV.,  was  defined  to  contain 
8  gallons.  It  thus  contains  80  pounds  of  distilled  water,  or 
2218.192  cubic  inches.  The  "heaped  bushel"  of  2815  cubic 
inches,  declared  by  the  same  act  as  a  measure  for  coals,  lime, 
potatoes,  fruit,  and  fish,  was  abolished  in  1835,  by  an  act 
of  Parliament,  during  the  reign  of  William  IV.  The  Win- 
chester bushel,  in  use  from  the  time  of  Henry  VII.  to  1826, 
contained  2150.42  cubic  inches.  The  original  standard  Win- 
chester bushel,  as  well  as  the  yard,  is  still  preserved  in  the 
museum  at  Winchester. 

The  unit  of  dry  measure  in  the  United  States  is  the  Win- 
chester bushel,  the  same  as  the  old  English  standard.  Its  form 
is  a  cylinder,  18^  inches  in  diameter,  and  8  inches  deep.     Its 


MEASURES   OF   EXTENSION.  511 

volume  is  2150.42  cubic  inches,  and  it  contains  77.627413 
pounds  avoirdupois  of  distilled  water,  at  its  maximum  density, 
39.8°  F.,  barometer  30  inches.  The  New  York  bushel  is  de- 
clared to  contain  80  pounds  of  distilled  water  at  its  maximum 
density,  and  is  thus  identical  with  the  Imperial  bushel  of  Great 
Britain.  The  chaldron,  consisting  in  some  places  of  36  bushels, 
and  in  others  of  32  bushels,  is  used  in  some  parts  of  the  United 
States  for  measuring  coal  and  coke,  but  is  being  discontinued 
here  as  it  has  been  in  England.  One  half  of  a  peck,  or  four 
quarts,  is  called  a  dry  gallon.  The  chaldron  was  divided  into 
vats,  sacks,  and  bushels.  The  coal  bushel  held  1  quart  more 
than  the  Winchester  bushel.  Twenty-one  chaldrons  made  s 
score. 

The  cental  is  a  standard  recently  recommended  by  the  Boards 
of  Trade  in  New  York,  Cincinnati,  Chicago,  and  other  large 
cities,  for  estimating  grain,  seeds,  etc.  Were  this  standard 
generally  adopted,  the  discrepancies  of  the  present  system  of 
grain  dealing  would  be  avoided.  Bushels  are  changed  to  cen- 
tals, by  multiplying  by  the  number  of  pounds  in  one  bushel, 
and  dividing  the  product  by  100.  The  remainder  will  be  hun- 
dredths of  a  cental. 


CHAPTER  III. 

MEASURES   OP   WEIGHT. 

WEIGHT  is  the  measure  of  the  force  of  gravity.  All  bodies 
are  attracted  towards  the  center  of  the  earth  in  propor- 
tion to  the  quantity  of  matter  contained.  This  influence  being 
a  constant  force,  it  was  seen  that  it  might  be  employed  in  com- 
paring bodies,  and  determining  their  relative  quantity  of  mat- 
ter. Some  standard  being  fixed  upon,  the  relation  of  bodies  to 
this  standard  may  be  expressed  numerically,  and  thus  there 
will  arise  a  system  of  measures  denominated  weights. 

Measures  of  Weight,  like  those  of  length,  originated  with 
natural  objects.  Seeds  seem  to  have  supplied  the  original  unit. 
The  carat,  used  for  weighing  in  India,  is  a  small  bean.  The 
basis  of  the  English  scale  of  weights  is  a  grain  of  wheat. 
Henry  III.  enacted  that  an  ounce  should  be  the  weight  of  640 
dry  grains  of  wheat  from  the  middle  of  the  ear.  The  penny- 
weight was  the  weight  of  an  English  penny,  which  varied  in 
size  until  the  time  of  Queen  Elizabeth. 

The  weight  of  a  body  is  the  measure  of  the  force  by  which 
it  is  drawn  towards  the  center  of  the  earth.  The  determination 
of  weight  consists  in  the  comparison  of  the  object  to  be  weighed 
with  some  fixed  standard.  Such  a  standard  could  not  be  pre- 
cisely defined  by  written  law  or  oral  explanation,  or  in  any 
way  except  by  the  test  of  muscular  resistance.  Having  a  fixed 
standard,  the  weight  of  bodies  would  be  estimated  by  comparing 
them  with  this  standard.  The  comparison  of  weight  was  not 
quite  so  readily  made  as  that  of  length,  as  a  balance  was  neces- 
sary, the  construction  of  which  required  some  degree  of  me- 

(512) 


MEASURES   OF   WEIGHT.  515 

chanical  knowledge.  The  balance,  or  scales,  however,  is 
known  to  have  been  in  existence,  in  a  rude  form,  from  very 
early  times.  The  Greeks,  as  appears  from  the  Parian  chroni- 
cle, believed  weights,  measures,  and  the  stamping  of  gold  and 
silver  coins  to  have  been  alike  the  invention  of  Phidon,  ruler  of 
Argos,  about  the  middle  of  the  8th  century  B.  C. 

The  standards  of  weight  were  even  less  definite  than  those 
of  length.  This  is  apparent  from  the  use  of  such  units  as 
stone,  load,  last,  etc.  Even  the  term  pound  (pondus)  implied 
only  weight  indefinitely.  The  grain,  taken  from  the  graius  or 
corns  of  wheat,  was,  as  a  standard  of  small  weights,  about  the 
only  denomination  of  weight  that  would  universally  convey 
anything  like  a  definite  idea.  A  statute  of  Henry  III.,  in  1266, 
enacts  "that  an  English  penny,  called  the  sterling,  round  with- 
out clipping,  shall  weigh  32  grains  of  wheat,  well  dried  and 
gathered  out  of  the  middle  of  the  ear;  and  20  pence  (penny- 
weights) to  make  an  ounce,  12  ounces  a  pound,  8  pounds  a 
gallon  of  wine,  and  8  gallons  of  wine  a  bushel  of  London, 
^vhich  is  the  8th  part  of  a  quarter." 

In  some  countries,  measures  of  weight  seem  to  have  had 
their  origin  in  the  measures  of  value.  Thus,  in  Latin,  to  pay 
money  was  to  weigh  it;  and  nearly  all  weights  are  supposed 
to  have  had  their  origin  in  the  practice  of  weighing  specie. 
The  Sicilian  pound,  which  has  been  adopted  in  all  countries 
that  accepted  the  Roman  system,  and  the  German  mark, 
both  fundamental  in  all  European  weights,  are  said  to  have 
been  originally  quantities  of  silver.  England  accepted  the 
pound  thus  derived,  and  also  attempted  to  give  precision  to 
smaller  sums  of  money  by  weighing  them  by  grains  of  corn,  as 
already  stated.  Even  as  late  as  the  time  of  Elizabeth,  pay 
ments  seem  to  have  been  invariably  made  by  weight;  indeed, 
the  whole  metrical  system  of  England  was  derived  from  money 
weights.  The  payments  by  tale  having  superseded  those  by 
weight,  the  real  origin  of  weights  is  liable  to  be  overlooked. 

As  science  was  developed,  it  began  to  concern  itself  in  the 
33 


514  THE   PHILOSOPHY   OP    ARITHMETIC. 

establishment  of  a  fixed  system  of  weights.  The  object  was  to 
find  some  invariable  standard  by  which  such  a  system  could 
be  derived.  As  there  is  a  constant  ratio  between  the  volumes 
and  weights  of  the  same  substances  when  placed  in  the  same 
physical  circumstances,  it  was  seen  that  standards  of  weight 
may  be  derived  from  those  of  length.  For  example,  a  cubic 
inch  of  distilled  water,  at  the  same  temperature  and  under  the 
same  atmospheric  pressure,  will  always  have  the  same  weight. 
Advantage  has  been  taken  of  this  property  of  bodies  to  connect 
measures  of  weight  with  those  of  length,  and  the  weight  of  a 
given  bulk  of  water  at  a  fixed  temperature  is  now  the  standard 
from  which  all  weights  are  derived. 

The  establishment  of  a  system  of  weights  early  engaged  the 
attention  of  the  English  people.  In  order  to  arrive  at  some 
definite  standard,  it  was  declared,  in  the  Great  Charter,  that 
the  weights  should  be  the  same  all  over  England;  but  no  ordi- 
nance, perhaps,  was  ever  so  ill  observed.  The  old  English 
pound,  which  is  said  to  have  been  the  legal  standard  of  weight 
from  the  time  of  William  the  Conqueror  to  that  of  Henry  VII., 
was  derived  from  the  weight  of  grains  of  wheat,  as  stated  above. 
Henry  VII.  altered  this  weight,  and  introduced  the  Troy 
pound  instead,  which  was  one-sixteenth  part,  or  three-fourths 
of  an  ounce,  heavier  than  the  Saxon  pound.  The  Troy  pound 
was  divided  in  the  same  manner  as  the  Saxon  pound,  into 
ounces,  pennyweights,  and  grains;  but  the  pennyweight  con- 
tained only  24  grains,  and  consequently  a  grain  Troy  became 
a  much  heavier  weight  than  the  grain  of  wheat.  In  fact,  the 
pound  Troy  contains  5,7G0  grains,  while  the  Saxon  pound, 
which  was  divided  into  7,680  grains,  contained  only  5,400 
Troy  grains.  The  avoirdupois  pound  was  introduced  by 
a  statute  of  Henry  VIII.  Its  first  object  was  to  weigh 
butchers'  meat  in  the  market,  but  it  gradually  came  to  be 
used  for  all  kinds  of  coarse  goods  or  merchandise.  Two 
legal  measures  of  weight  were  thus  established,  and  have  con- 
tinued to  be  used  in  England  ever  since,  and  have  been  also 


MEASURES   OF   WEIGHT.  51l» 

introduced  into  this  country.  The  standard  of  these  weights 
was  definitely  fixed  by  Act  of  Parliament  in  1824.  The 
standard  brass  weight  of  the  pound  Troy,  made  in  1758,  and 
then  in  custody  of  the  Clerk  of  the  House  of  Commons,  was 
declared  to  be  the  Imperial  standard  Troy  pound,  and  that 
7,000  Troy  grains  shall  constitute  an  avoirdupois  pound. 

The  retention  of  two  different  systems  of  weights  was  in 
compliance  with  the  common  usages  of  the  country.  Mr. 
Davies  Gilbert  stated  the  reasons  for  it  as  follows:  "The  Troy 
pound  appeared  to  us  to  be  the  ancient  weight  of  this  kingdom, 
having,  as  we  have  reason  to  suppose,  existed  in  the  same  state 
in  the  time  of  Edward  the  Confessor;  and  there  are  reasons, 
moreover,  to  believe  that  the  word  Troy  has  no  reference  to 
any  town  in  France,  but  rather  to  the  monkish  name  given  to 
London  of  Troy  Novant,  founded  on  the  legend  of  Brute. 
Troy  weight,  therefore,  according  to  this  etymology,  in  fact,  is 
London  weight." 

It  was  also  enacted  that  if  the  standard  Troy  pound  should 
be  lost  or  destroyed,  it  was  to  be  restored  by  a  reference  to  a 
cubic  inch  of  distilled  water,  which  has  been  found  and  is  de- 
clared to  be  252.458  Troy  grains  at  the  temperature  of  62° 
Fahrenheit,  the  barometer  being  at  30  inches.  The  weight  of 
a  pennyweight  Troy  is  thus  to  that  of  a  cubic  inch  of  distilled 
water  in  such  circumstances,  as  24  to  252.458,  or  of  24,000  to 
252,458;  so  that  the  weight  of  the  cubic  inch  of  distilled  water 
must  be  conceived  to  be  divided  into  252,458  equal  parts,  and 
24,000  of  such  parts  will  be  the  standard  pennyweight,  or  240 
of  such  pennyweights  will  be  the  standard  pound. 

A  committee  appointed  in  1843  published  a  report  in  1854, 
in  which  they  determined  to  take  the  avoirdupois  pound  of 
7,000  grains  as  the  standard,  and  to  construct  the  Troy  pound 
from  it.  They  took  the  brass  Troy  pounds  in  the  custody 
of  the  Exchequer  and  certain  platinum  pounds  in  the  possession 
o(  the  Royal  Society  and  others;  but  the  former  bein<j  found  to 
have  gained  in  weight  by  oxidation,  only  the  platinum  pounds 


516  THE    PHILOSOPHY    OF    ARITHMETIC. 

were  used.  From  them  a  platinum  pound  was  prepared, 
weighing  in  vacuo  6999.99345  grains.  The  form  of  the  weight 
is  a  cylinder,  with  a  groove  surrounding  it  a  little  above  the 
middle  of  its  height,  for  the  insertion  of  the  ivory  fork  used  in 
lifting  it.  The  weight  is  enclosed  in  a  mahogauy  box,  the  parts 
of  which,  when  screwed  together,  cause  the  weight  to  be  im- 
movable. This  box  is  enclosed  in  another  box,  and  with  the 
standard  yard  in  a  third  box,  and  finally  in  a  stone  case,  in  the 
vaulted  stone  room  of  the  Exchequer. 

For  philosophical  purposes  and  in  delicate  weighing,  Troy 
weight  only  is  used,  and  the  weight  is  usually  reckoned  in 
grains  liy  this  means  fractional  numbers  are  avoided,  and  no 
ambiguity  can  arise,  as  there  are  no  other  grains  than  Troy 
grains.  Dr.  Kelly,  in  his  Universal  Cambist,  an  elaborate  and 
useful  work,  states  that  the  dram  avoirdupois,  like  the  drachm 
of  the  apothecaries,  has  sometimes  been  divided  into  3  scruples 
and  60  grains;  but  as  no  such  weight  as  an  avoirdupois  grain 
ever  existed,  the  use  of  the  expression  is  an  instance  of  the 
confusion  inseparable  from  having  different  systems  of  weights 
n  which  the  same  names  are  applied  to  things  totally  distinct. 

Aside  from  the  British  statute  weights,  there  are  in  England 
numerous  other  discordant  denominations  of  weight,  used  for 
weighing  different  kinds  of  merchandise.  One  of  the  most 
common  of  these  is  the  stone,  which  has  a  great  variety  of 
different  significations.  In  Loudon,  however,  only  two  stones 
are  generally  used,  the  one  of  8  pounds  for  butchers'  meat,  and 
another  of  14  pounds,  for  other  commodities.  A  stone  of  glass 
has  been  reckoned  at  5  pounds.  The  following  are  some  of  the 
other  old  weights:  A  seam  of  glass,  equal  to  24  stones  ;  a  truss 
of  hay,  equal  to  56  pounds;  a  truss  of  new  hay  until  the  first 
of  September,  equal  to  60  pounds ;  a  truss  of  straw,  equal  to 
36  pounds.  In  weighing  wool  the  following  denominations  have 
been  used:  7  pounds  equal  1  clove ;  2  cloves  equal  1  stone;  2 
stones  equal  1  tod;  6j  tods  equal  1  wet/;  2  weys  equal  1 
sack  ;  12  sacks  equal  1  last  ;  a  pack  of  wool  equals  240  pounds. 


MEASURES   OF   WEIGHT.  517 

Tn  weighing  cheese  and  butter,  8  pounds  equal  1  clove,  and 
56  pounds  equal  1  tirkin.  Many  of  these  weights  are  now  ob- 
solete. 

Ancient  English  Measures. — The  basis  of  the  ancient  English 
measures  of  capacity  and  weight  was  the  ancient  Anglo-Saxon 
pouud.  This  pound  contained  5,400  grains,  the  grains  beiu^* 
22£  to  the  pennyweight,  and  the  pennyweight  being  equal  to 
32  grains  of  average  quality  taken  from  the  middle  of  the  ear. 
Eight  of  these  pounds  formed  the  gallon  of  dry  and  liquid 
measure,  8  of  these  gallons  the  bushel,  and  8  of  these  bushels 
the  quarter.  The  old  English  pound  stood  to  the  Troy  pouud 
as  15  to  16.  A  few  of  these  standard  gallons  and  bushels  have 
been  preserved,  and  are  found  to  be  somewhat  less  than  the 
proportion  indicated.  The  Troy  pound  was  not  known  as  a 
legal  standard  until  certain  changes  were  introduced  into  the 
currency  by  Henry  VIII. 

The  sack  of  wool  was  roughly  reckoned  as  equal  to  the  quar- 
ter of  corn;  15  Saxon  ounces  formed  the  libra  mercaloria,  or 
pound  of  7000  grains  now  called  avoirdupois.  Fourteen  such 
pounds  made  the  stone  of  wool,  and  28  such  stones  constituted 
the  sack.  This  calculation,  however,  makes  the  sack  of  wool 
lighter  than  the  quarter  of  wheat  by  nearly  four  pounds. 

Another  ancient  weight  was  the  charrus  of  lead.  It  con- 
tained 2,100  avoirdupois  pounds;  and,  divided  by  the  old  hun- 
dred, 108  pounds,  is  found  to  contain  nearly  19^  hundred,  which 
is  the  modern  /other  or  fodder.  The  charrus  contained  30 
fotmale,  or  pedes,  each  pes  containing  6  stone,  less  2  pounds 
The  foot  or  pig  of  lead  is  the  tenth  of  a  cubic  foot  of  lead. 
Iron  was  measured  by  the  piece,  25  of  which  formed  the  hun- 
dred weight  of  108  pounds.  Wax  and  spices  were  estimated 
by  the  same  hundred.  A  last  of  wool  was  12  sacks;  a  last  of 
herrings,  ten  thousand,  each  hundred  being  120  ;  a  last  of  hides 
was  100,  that  is  10  dakers  or  dikers,  each  diker  being  ten. 

The  American  System. — Our  system  of  weights  was  derived 
from  those  of  the  mother  country.     These,  as  already  shown, 


518  THE   PHILOSOPHY   OF    ARITHMETIC. 

were  not  framed  by  scientific  men,  but  assumed  their  present 
form  gradually,  influenced  by  various  circumstances,  from  which 
cause  arose  that  irregularity  which  they  exhibit.  There  are 
four  kinds  of  weight  in  common  use  •  Troy  Weight,  Apotheca- 
ries' Weight,  Avoirdupois  Weight,  and  Diamond  Weight. 

Troy  Weight. — Troy  Weight  is  Used  for  weighing  gold,  sil- 
ver, jewels,  in  ascertaining  the  strength  of  liquors,  in  philosoph- 
ical experiments,  etc.  The  term  Troy  is  said  to  be  derived  from 
Troyes,  the  name  of  a  town  in  France,  where  the  weight  was 
first  used  in  Europe,  it  having  been  brought  from  Cairo  in  Egypt 
during  the  Crusades  of  the  12th  century.  Others  maintain 
that  it  has  no  reference  to  any  town  in  France,  but  rather  to 
the  monkish  name  given  to  London,  of  Troy  Novant,  founded 
on  the  legend  of  Brute,  Troy  weight  being,  therefore,  London 
weight. 

The  term  pound  is  from  the  Latin  pendo,  I  bend  or 
weigh.  The  term  ounce  is  from  the  Latin  uncia,  a  twelfth  part, 
the  ounce  being  one-twelfth  of  a  pound.  The  pennyweight  was 
the  weight  of  the  old  English  penny.  The  term  grain  is  from 
a  grain  of  wheat,  which  was  the  primitive  standard  of  all  the 
weights  in  England.  Thirty-two  of  these  taken  from  the  mid- 
dle of  the  ear  and  well  dried,  constituted  a  pennyweight,  20 
pennyweights  an  ounce,  and  12  ounces  a  pound.  The  pound 
thus  derived  was  the  legal  standard  of  weight  from  the  time  of 
William  the  Conqueror  to  that  of  Henry  VII.  The  latter 
king  changed  this  weight  and  introduced  the  Troy  pound,  which 
was  Jg-  part,  or  J  of  an  ounce,  heavier  than  the  Saxon  pound. 
The  Troy  pound  was  divided  in  the  same  manner  as  the  Saxon 
pound,  that  is,  into  ounces,  pennyweights,  and  grains;  but  the 
pennyweight  contained  only  24  grains,  and  consequently  a 
grain  Troy  became  much  heavier  than  a  grain  of  wheat. 

The  Troy  pound  is  the  standard  unit  of  weight  in  the  United 
States,  and  is  the  same  as  the  Imperial  pound  Troy  of  Great 
Britain.  It  is  equal  to  the  weight  of  22.794377  cubic  inches 
of  distilled  water,  at  the  temperature  of  39.83°  Fahrenheit,. 


MEASURES   OF   WEIGHT.  519 

the  barometer  at  30  inches.  In  the  United  States  Mint,  the 
Troy  ounce  is  adopted  as  the  standard,  and  all  weights  are  ex- 
pressed in  decimal  multiples  and  sub-multiples  of  the  ounce. 

The  symbol  oz.  is  from  the  Spanish  word  onza,  signifying 
ounce,  though  Webster  derives  it  from  the  character  3,  placed 
afier  the  0,  according  to  an  ancient  method  of  abbreviating  ter- 
minations; lb.  is  from  the  Latin  libra,  a  pound;  pwt.  is  a  com- 
bination of  p.  for  penny,  and  wt.  for  weight ;  dwt.,  from  dena- 
rius and  weight,  is  nearly  obsolete,  and  seems  a  less  appro- 
priate term  than  pwt.,  being  partly  Latin  and  partly  English. 
Apothecaries'  Weight. — Apothecaries'  Weight  is  used  only  in 
mixing  medicines.  Apothecaries  buy  and  sell  their  drugs  by 
Avoirdupois  Weight.  The  name  arises  from  the  weight  being 
used  by  apothecaries.  The  term  scrapie  is  from  the  Latin 
scrupulus,  a  little  stone.  The  term  dram  is  from  the  Greek 
drachma,  a  piece  of  money. 

The  symbols  have  been  supposed  to  be  modifications  of  the 
figure  3,  suggested  by  there  being  3  scruples  in  a  dram.  Cham- 
pollion,  however,  has  traced  them  back  to  the  hieroglyphics  of 
Egypt.  The  unit  is  the  pound,  and  is  identical  with  the  Troy 
pound,  as  are  also  the  ounce  and  grain,  the  ounce  being  differ- 
ently divided. 

Avoirdupois  Weight. — Avoirdupois  Weight  is  used  for 
weighing  everything  except  jewels,  gold,  silver,  liquors  in  phil- 
osophical experiments,  etc.  Avoirdupois  Weight,  it  is  said, 
was  introduced  by  a  statute  of  Henry  VIII.  The  term  occurs 
in  some  orders  of  his,  A.  D.  1532  ;  and  Queen  Elizabeth,  in  1588, 
ordered  a  pound  of  this  weight  to  be  deposited  in  the  Exche- 
quer as  a  standard.  It  was  first  used  in  England  to  weigh 
butchers'  meat,  but  gradually  came  to  be  used  to  weigh  all 
kinds  of  coarse  goods  or  merchandise. 

The  term  Avoirdupois  is  said  to  be  derived  from  the  French 
avoir  du  poidst  signifying  to  have  weight.  Others  think  it  is 
from  avoirs,  the  ancient  name  of  goods  or  chattels,  and  poids, 
signifying  weight  in  the  Norman  dialect.     Another  authority 


520  THE    PHILOSOPHY    OF    ARITHMETIC. 

says  it  is  from  the  old  French  aver  de  pes,  property  or  mer- 
chandise of  weight,  translated  by  Kelham,  "any  bulky  com- 
modities." Still  another  derivation  is  from  the  old  French 
verb  averer,  to  verify.  The  term  ton  is  from  the  Saxon  tunne, 
a  cask.  The  origin  of  the  other  terms  has  already  been  given. 
The  symbol  cwt  is  from  centum  and  weight. 

The  unit  is  the  pound.  It  consists  of  7000  Troy  grains,  and 
is  consequently  heavier  than  the  pound  Troy,  which  contains 
only  5,760.  The  ounce  Avoirdupois,  however,  is  lighter  than 
the  ounce  Troy,  owing  to  the  difference  in  the  division  of  the 
pound.  A  pound  of  feathers  is  therefore  heavier  than  a  pound 
of  gold,  while  an  ounce  of  gold  is  heavier  than  an  ounce  of 
lead. 

The  standard  Avoirdupois  pound  of  this  country  is  the 
weight  of  27-7015  cubic  inches  of  distilled  water,  at  its  maxi- 
mum density,  or  39.83°  Fahrenheit,  weighed  in  the  air,  the 
barometer  being  at  30  inches.  It  is  identical  with  the  Imperial 
pound  Avoirdupois  of  Great  Britain,  which  is  the  weight  of 
27.7274  cubic  inches  of  distilled  water  at  the  temperature 
of  62°  Fahrenheit.  The  difference  in  the  number  of  cubic 
inches  is  owing  to  the  difference  of  temperature  of  the  water 
employed. 

In  Great  Britain  28  pounds  equal  1  quarter,  112  pounds 
equal  1  hundredweight,  and  2240  pounds  equal  1  ton.  These 
are  called  the  long  hundred  and  long  ton ;  they  were  formerly 
used  in  this  country,  but  are  now  only  used  at  the  custom- 
houses in  invoices  of  English  goods,  in  the  wholesale  iron 
and  plate  trade,  and  in  wholesaling  and  freighting  coal  from 
the  coal  mines  of  Pennsylvania. 

Diamond  Weight. — Diamond  Weight  is  used  in  weighing 
diamonds  and  other  precious  stones.  In  this  weight,  16  parts 
equal  1  grain,  and  4  grains  equal  1  carat.  One  grain  of 
this  weight  equals  i  of  a  grain  Troy.  The  term  carat  is 
also  used  to  indicate  a  proportional  part  of  a  given  weight, 
and  is  then  called  asxay  carat.  Each  assay  carat  consists  of 
4  assay  grains,  and  each  assay  grain,  of  4  assay  quarters. 


CHAPTER  IV. 

MEASURES   OF   VALUE. 


rllE  Value  of  anything  is  its  worth,  or  it  is  the  property  or 
properties  of  a  thing  which  render  it  useful  or  estimable. 
It  has  also  been  defined  as  the  estimate  of  a  given  commodity  in 
comparison  with  other  commodities.  It  is  readily  seen  that  it 
is  not  easy  to  give  a  definition  of  value  entirely  satisfactory. 
The  two  principal  elements  of  the  valne  of  anything,  are  utility 
and  difficulty  of  attainment. 

Value  is  distinguished  from  price,  which  is  the  estimate 
given  of  any  commodity  by  one  value  alone — the  value  of  the 
precious  metals.  There  can  be  a  general  rise  in  prices,  since 
the  value  of  the  single  measure  may  fall;  but  there  cannot  be 
a  general  rise  in  values,  for  values  are  relative  and  mutual. 
There  may  be,  of  course,  a  great  rise  or  fall  in  any  one  thing 
when  it  is  scarce  and  in  demand,  or  abundant  and  neglected. 
There  cannot  be  a  universal  rise  or  fall  in  values,  for  if  such  a 
case  could  be  conceived,  no  one  would  get  any  more  or  less  in 
either  case  than  before. 

The  value  of  any  commodity  or  service  is  affected  by  two 
causes— demand,  which  is  temporary;  and  the  cost  of  produc- 
tion, which  is  permanent.  In  the  long  run,  a  particular  value 
conforms  to  the  last  named  cause;  but  from  time  to  time  it  is 
regulated  by  demand  and  supply,  and  may,  therefore,  rise  and 
fall,  far  above  or  far  below  the  cost  of  production.  From  the 
fact  that  demand  and  labor  determine  values,  some  economists 
have  insisted  that  the  first  of  these  causes  is  the  true  measure 
of  values  ;  while  others  maintain  that  value  is  determined  by 

(521  ) 


522  THE    PHILOSOPHY    OF    ARITHMETIC. 

the  second.  Both,  however,  are  determining  causes,  though 
under  different  circumstances;  and  the  statements,  that  labor 
is  the  cause  of  value,  and  demand  is  the  cause  of  value,  are, 
though  apparently  contradictory,  two  phases  of  the  same  fact. 

A  measure  of  value  is  one  of  the  primal  necessities  of  society. 
The  united  dependence  of  individuals  creates  the  necessity  of 
an  exchange  between  two  services  or  utilities.  For  such  an  act 
of  exchange,  it  is  essential  that  the  thiugs  exchanged  should  be 
measured  by  some  standard  of  value  well  understood  between 
the  contracting  parties.  To  illustrate,  suppose  A  produces 
shoes  and  B  produces  bread.  Now  A  may  want  bread  before 
B  wants  shoes.  The  immediate  exchange  of  shoes  for  bread 
would  not  be  convenient;  but  the  exchange  between  the  parties 
may  still  take  place  by  the  intervention  of  some  medium  of 
exchange.  If  B  receives  this  medium  for  bread,  he  takes  it  in 
the  faith  that  at  bis  pleasure  he  may  complete  the  exchange 
with  A  by  the  purchase  of  shoes,  or  may  even  employ  the  right 
to  shoes  assigned  to  him  by  the  transfer  of  a  portion  of  this 
medium,  in  procuring  any  other  utility  which  he  may  desire. 
Hence  a  sale  is  said  to  be  half  of  an  exchange. 

The  necessity  of  some  simple  measure  or  representative  of 
value,  which  could  be  employed  as  a  medium  of  exchange,  would 
be  early  felt.  The  objects  first  selected  seem  to  have  been  organ- 
ized bodies.  Thus  cattle,  pigs,  dried  fish,  and  the  skins  of 
animals,  have  been  used.  The  tendency,  however,  seems  to 
have  been  to  employ  metallic  substances.  At  first  the  baser 
metals  were  used  as  money.  Iron  was  the  primitive  money 
of  the  Lacedemonians,  and  copper  of  the  Romans.  In  most 
countries  the  precious  metals  were  early  adopted  for  this  pur- 
pose. When  first  used,  they  were  in  the  shape  of  bars  or 
ingots,  and  were  exchanged  by  weight.  Aristotle  and  Pliny 
tell  us  that  this  is  the  method  by  which  the  precious  metals 
were  originally  exchanged  for  other  things  in  Greece  and 
Italy.  The  Bible  also  states  that  Abraham  weighed  200  shekels 
of  silver  and  gave  them  in  exchange  for  a  piece  of  ground 
which  he  had  purchased  from  the  sons  of  Heth. 


MEASURES   OF    VALUE.  523 

The  medium  of  exchange,  it  is  clear,  must  be,  so  far  as  possi 
ble,  of  persistent  value,  that  is,  liable  to  the  fewest  possible 
fluctuations  of  intrinsic  value,  and  capable  of  being  transferred 
for  very  nearly  equal  quantities  of  utilities  at  deferred  periods. 
Hence  the  basis  of  a  circulating  medium  must  be  a  commodity 
which  is  of  nearly  absolute  value.  To  possess  this  quality,  it 
must  be  produced  in  nearly  equal  quantities  by  very  nearly 
equal  labor.  Almost  the  only  things  which  possess  these 
characteristics  are  the  precious  metals,  gold  and  silver;  and 
consequently  they  have  been  almost  universally  selected  as  a 
medium  of  exchange.  These  two  metals  are  still  further 
adapted  to  a  monetary  use  by  being  comparatively  indestructi- 
ble. Something  is  needed  that  can  be  treasured  up  for  an  in- 
definite period,  without  spontaneous  alteration,  waste,  or 
decomposition,  while  waiting  for  an  exchange.  The  material 
selected  should  also  be  homogeneous,  or  of  equal  value  through- 
out its  whole  substance;  and  further,  it  must  also  be  susceptible 
of  easy  division  ard  reunion.  Since  these  qualities  are  pos- 
sessed almost  exclusively  by  gold  and  silver,  it  is  not  surprising 
that  all  societies,  spontaneously  and  as  by  instinct,  have  adopted 
the  precious  metals  as  money. 

There  is  also  a  peculiar  fitness  found  in  the  relation  of  gold 
and  silver  to  each  other,  giving  us  two  media  of  different  fixed 
values.  The  circumstances  under  which  they  are  found,  and 
the  labor  required  to  produce  them,  are  such  that  there  has 
been  but  little  disturbance  in  the  mutual  values  of  gold  and 
silver;  and  although  there  can  never  be  a  precise  ratio  of  in 
trinsic  value  possessed  by  each  of  the  two  metals,  yet  in  modern 
times  the  margin  of  oscillation  is  so  narrow,  that  both  may  be 
used  simultaneously  as  media  of  exchange,  the  one  for  larger 
and  the  other  for  smaller  values. 

In  order  to  answer  for  money,  however,  a  mass  of  metal 
must  be  issued  by  an  authority  which  gives  to  it  a  practical 
guaranty  of  its  weight  and  fineness.  It  has  thus  been  neces- 
sary for  the  government  of  every  civilized  country  to  coin  its 


624:  THE    PHILOSOPHY    OF    ARITHMETIC. 

own  money,  to  prevent  the  coinage  by  private  parties,  and  to 
prohibit  by  severe  penalties  the  forging  of  coins,  the  fabrica- 
tion of  counterfeit  coins,  or  coins  of  less  weight  than  the  stan- 
dard, or  made  up  in  whole  or  in  part  of  some  baser  or  less 
valuable  metal.  The  necessity  for  this  security  is  so  great  that 
however  much  governments  have  tampered  with  the  weight  of 
the  nominal  quantity,  they  have  seldom,  unless  thoroughly  de- 
moralized and  desperate,  ventured  on  altering  or  debasing  the 
standard;  and  when  they  have  done  so,  the  result  has  been 
ruinous  in  the  last  degree. 

It  was  the  practice  in  early  ages  to  pay  money  by  weight ; 
from  which  it  would  seem  that  coins,  in  the  strict  sense  of 
metallic  masses  of  a  certified  weight,  were  unknown.  The 
practice  of  weighing  money,  however,  continued  for  ages  after 
coins  were  in  use.  Where  the  system  of  coinage  originated  is 
not  known,  though  it  has  been  ascribed  to  different  persons. 

He  who  first  shaped  a  metal  into  pieces  of  convenient 
size,  marked  with  a  distinct  value,  thus  avoiding  the  need  of 
the  hammer  and  chisel  to  cut  it  off,  and  a  balance  to  weigh  it, 
was  the  first  inventor  of  coins.  History  is  silent  respecting 
his  name,  his  country,  and  the  date  of  the  invention.  Homer 
speaks  of  workers  in  metals,  but  makes  no  mention  of  coined 
money.  Herodotus  says  the  Lydians,  so  far  as  he  knew,  were 
the  first  to  use  struck  money,  and  there  are  reasons  for  thinking 
with  him  that  the  invention  was  Asiatic.  The  subject  will  be 
more  fully  considered  subsequently. 

Originally  the  coins  of  all  countries  seem  to  have  had  the 
same  denominations  as  the  weights  commonly  used  in  them, 
and  contained  the  exact  quantity  of  precious  metals  indicated 
by  their  name.  Thus,  the  talent  was  a  weight  used  in  the 
earliest  period  by  the  Greeks,  the  as  or  pondo  by  the  Romans, 
the  livre  by  the  French,  and  the  pound  by  the  English  and 
Scotch  ;  and  the  coins  originally  in  use  in  Greece,  Italy,  France, 
and  England,  bore  the  same  names  and  weighed  precisely  a 
talent,  a  pondo,  a  livre,  and  a  pound.     This  arose  from  the 


MEASURES    OF    VALUE.  525 

original  custom  of  making  payments  by  the  weight  of  the  arti- 
cles used  as  a  medium  of  exchange.  The  standard  has  not, 
however,  been  preserved  inviolate,  in  either  ancient  or  modern 
times.  It  has  been  less  degraded  in  England  than  anywhere 
else;  but  even  there,  the  quantity  of  s.lver  in  a  pound  sterling 
is  less  than  one-third  part  of  a  pound  weight.  In  France,  the 
livre  current  in  1789  contained  less  than  one-sixty-sixth  part 
of  the  silver  implied  in  its  name.  In  Spain  and  some  other 
countries  the  depreciation  has  been  carried  still  further. 

When  the  use  of  coins  has  once  been  adopted,  all  values  in  con- 
tracts and  other  engagements  are  rated  or  estimated  in  money, 
and  it  is  usual  in  almost  all  countries  to  enact  that  coins  of 
legal  and  standard  weight  and  purity  shall  be  legal  tender,  and 
to  declare  that  no  legal  proceedings  of  any  kind  shall  be  insti- 
tuted on  account  of  any  debt  or  pecuniary  obligation  against 
any  individual  who  has  offered  to  liquidate  the  same  by  pay- 
ment of  an  equivalent  amount  of  the  recognized  coin  of  the 
country. 

In  the  use  of  metals  for  money,  it  should  be  remembered 
there  is  simply  an  exchange  of  values.  Equivalents  are  still 
given  for  equivalents.  The  exchange  of  a  barrel  of  flour  for 
an  ounce  of  unfashioncd  gold  is  as  much  a  barter  as  if  it  were 
exchanged  for  an  ox  or  a  barrel  of  beer;  and  if  the  metal  were 
formed  into  a  coin,  or  marked  with  a  stamp  declaring  its  weight 
and  fineness,  it  would  make  no  difference  in  the  nature  of  the 
exchange.  The  notion  has  been  entertained  that  coins  are 
merely  the  signs  of  values.  But  they  have  no  more  claim  to 
this  designation  than  bars  of  iron  or  copper,  sacks  of  wheat,  or 
any  other  article.  A  draft  or  check  may  not  improperly  be  re- 
garded as  a  symbol  of  value  ;  but  a  coin  is  itself  an  article  of 
value.     A  dollar  is  not  a  sign  ;  it  is  the  thing  signified. 

The  use  of  two  precious  metals,  gold  and  silver,  gives  what 
is  called  a  double  currency.  This  produces  some  inconvenience 
on  account  of  slight  variations  in  the  relative  value  of  the  two 
metals.     In  England,  in  1803,  the  proportion  was  fixed  at  1  to 


526  THE    PHILOSOPHY    OF    ARITHMETIC. 

15£,  which  slightly  undervalued  the  customary  proportion  of 
gold,  and  hence  gold  was  seldom  or  never  seen.  The  great 
discoveries  of  gold  in  Australia  and  California  slightly  re- 
versed the  ratio,  undervaluing  the  silver,  the  consequence  of 
which  is  that  gold  has  been,  to  a  great  extent,  substituted  for 
silver,  and  the  latter  metal  exported  in  vast  quantities.  It 
seems  likely,  however,  on  account  of  improvements  in  methods 
of  extracting  silver  from  other  ores  and  the  cheapening  of 
quicksilver,  that  the  margin  of  oscillation  will  be  consider- 
ably narrowed.  Such  oscillations,  however,  it  is  apparent,  lead 
to  many  inconveniences  in  the  use  of  a  double  currency,  and 
France  is  said  to  be  the  only  nation  in  which  both  metals  are 
a  legal  tender.  In  England  and  our  own  country,  gold  is  the 
actual  standard.  Silver  and  copper  are  issued,  but  are  so 
much  overvalued  that  they  could  not  be  exported  in  the  shape 
of  coins,  and  yet  so  regulated  as  to  obviate  the  risk  of  pri- 
vate coining.  In  England,  silver  is  not  legal  tender  for  more 
than  40s.,  nor  copper  for  more  than  12d.,  or  if  offered  in  farth- 
ings, for  6d.  In  the  United  States,  silver  is  a  legal  tender  up 
to  $5.  The  German  Empire  has  adopted  gold  alone  as  a  legal 
tender;  and  Denmark,  Sweden,  the  Netherlands,  and  some  othei 
countries,  have  done  the  same. 

Paper  Money. — But  how  great  soever  the  advantages  result 
ing  from  the  employment  of  gold  and  silver  as  money,  there- 
are  also  many  disadvantages.  The  use  of  a  metallic  currency 
is  accompanied  by  a  heavy  expense ;  and  there  is  a  much  greater 
difficulty  in  effecting  payments  by  the  use  of  coin  than  we 
might  at  first  suppose.  The  cost  of  the  wear  and  tear  of  the 
currency  of  a  large  country  like  England  or  the  United  States, 
allowing  only  a  small  percentage  for  the  same,  would  amount 
to  several  millions  of  dollars  a  year.  But  the  difficulty  or  in- 
convenience of  the  transportation  of  coin  in  making  payments, 
is  even  a  more  serious  objection.  A  million  of  dollars  in  gold 
would  weigh  nearly  two  tons,  and  would  require  a  wagon  to 
transport  it.     It  is  also  inconvenient  to  make  small  payments 


MEASURES   OF    VALUE.  527 

between  places  remote  from  each  other,  inasmuch  as  the  expense 
of  sending  gold  by  express,  and  the  premium  to  guarantee  it 
against  loss,  amount  to  a  considerable  sum.  llence  arose  the 
necessity  of  using  for  money  some  less  valuable  and  more 
portable  material  than  bullion ;  and  hence,  also,  the  origin  of 
bills  of  exchange,  checks,  and  other  devices  for  economizing 
the  use  of  money.  The  importance  of  such  a  representative  cur- 
rency appears  also  in  the  fact  that  without  it  at  least  four  or 
five  times  as  much  gold  and  silver  would  be  needed  as  with  it. 
Indeed,  without  some  such  tokens  or  representatives  of  money, 
it  would  be  almost  impossible  to  meet  the  demands  of  barter 
and  commerce  in  civilized  countries. 

Of  the  substitutes  for  gold  and  silver,  paper  notes,  payable 
on  demand,  have  been  by  far  the  most  generally  adopted,  and 
are,  in  all  respects,  the  most  eligible.  Intrinsically  they  are 
almost  destitute  of  value,  so  that  their  employment  and  their 
loss  costs  next  to  nothing;  and  they  may  be  carried  about  or 
transmitted  by  mail  with  the  utmost  facility.  Possessing  no 
value  of  themselves,  their  worth  must  depend  entirely  on  arti- 
ficial means  or  regulations.  They  are  usually  issued  as  substi- 
tutes for,  or  a  representative  of  coin,  the  issuer  being  bound  to 
pay  in  coin  the  sums  they  represent,  on  demand  of  the  holder 
For  the  issue  of  such  notes  it  has  been  found  necessary  to  es- 
tablish banks  of  issue,  and  the  questions  that  grow  out  of  a 
banking  system  are  among  the  most  interesting  that  can  pre- 
sent themselves  to  the  mind  of  the  economist.  Bank  notes, 
however,  are  not  the  only  forms  of  a  representative  currency. 
Bankers'  checks,  private  checks,  bills  of  exchange,  etc.,  and  all 
analogous  securities,  accomplish  the  same  purpose. 

The  basis  of  such  securities,  it  must  be  remembered,  is 
confidence — the  confidence  of  the  holder  that  they  can  be 
converted  at  his  discretion  into  that  which  alone  fulfills  the 
conditions  of  a  currency,  the  precious  metals.  It  thus  appears 
that  the  most  of  the  business  of  civilized  countries  is  not 
transacted  with  money,  but  upon  faith,  confidence  in  promises, 


528  THE    PHILOSOPHY    OF    ARITHMETIC. 

a  belief  that  men  and  institutions  will  do  what  they  have 
promised.  Notes  are  sometimes  issued  expressed  in  money 
value,  but  based  on  other  values,  as  land,  shares  of  stock,  etc., 
and  no  harm  ensues  ordinarily,  as  no  one  will  take  them  for 
more  than  they  are  worth;  and  since  there  is  no  compulsion, 
society  will  accept  no  more  than  it  needs. 

An  essential  e-Iernent  of  such  a  currency  is  that  it  must  be 
convertible  and  voluntary.  Occasionally  governments  give 
their  own  paper,  or  the  paper  of  some  institution  under  their 
control,  a  forced  circulation.  Such  an  act  disturbs  the  currency 
of  a  country,  and  often  produces  suffering  and  lasting  injury  to 
credit.  A  convertible  currency  cannot  be  extended  beyond 
the  amount  which  a  community  requires;  but  an  inconvertible 
currency  may  be  issued  to  any  amount,  and  may  be  made  to 
circulate  extensively.  The  immediate  effect  of  such  an  issue 
is  to  displace  the  metallic  currency,  which  is  either  hoarded  or 
exported.  A  very  small  premium  on  the  precious  metals  is 
sufficient  to  banish  them  from  circulation ;  and  the  forced  currency 
being  over-valued  in  comparison  with  the  metallic  currency,  and 
of  compulsory  acceptance,  no  one  will  pay  in  the  dearer  medium ; 
and  the  coin  will  command  a  premium,  or  paper  will  be  at  a 
discount. 

Such  an  act  on  the  part  of  a  government  is  a  wrong  to  the 
citizen  as  well  as  a  source  of  disaster  to  the  state.  To  promise 
money  and  give  something  else,  is  a  fraud;  and  to  force  the 
acceptance  of  such  a  currency  is  as  great  a  robbery  of  the 
public  as  the  circulation  of  base  money.  The  very  fact  of  a 
circulation  being  compulsory  is  an  indication  that  the  currency 
is  not,  on  its  own  merits,  worth  its  nominal  value;  and  the 
result  is  that  such  securities  not  only  greatly  depreciate,  but 
often  become  worthless,  and  their  repudiation  inevitable. 
Among  the  most  notable  examples  of  such  a  currency  are  Law's 
bank  under  the  Regency,  the  South  Sea  bubble,  the  French 
assiguats,  and,  to  a  certain  extent,  the  colonial  currency  during 
the  American  Revolution.     It  may  be  stated  that  governments 


MEASURES   OF   VALUE.  52$ 

seldom  resort  to  such  an  act  except  in  the  exigencies  of  war  r 
and  it  always  requires  years  for  the  restoration  of  the  proper 
relations  between  the  paper  and  metallic  currency. 

History  of  Money. — Money  has  been  employed  as  a  medium 
of  exchange  from  the  very  earliest  historical  periods.  In  ancient 
Greece  and  Rome,  cattle  were  used,  from  which  we  derive  our 
word  pecuniary,  which  is  from  pecunia,  and  this  from  pecus, 
cattle.  In  early  Greece  there  was  a  currency  of  "  spits  "  or 
"skewers,"  six  of  which  made  a  drachm,  or  handful;  they 
were  probably  nails  of  iron  or  copper.  The  Lacedemonians 
and  others  used  iron  money.  Among  the  most  ancient  existing 
specimens  of  coin  are  those  of  electrum,  an  alloy  of  gold  with 
one-fifth  of  silver.  Gold,  silver,  and  copper  were  coined  by  the 
Greeks  and  Romans;  tin  was  coined  by  Dionysius  I.,  tyrant 
of  Syracuse,  and  Roman  and  British  tin  coins  are  known  to 
exist.  Early  leaden  money  is  mentioned;  a  leaden  stater  is 
preserved  in  the  British  museum,  and  leaden  money  is  now 
current  in  the  Burman  Empire.  Numa  Pompilius,  King  of 
Rome  about  TOO  B.  C,  made  money  of  both  wood  and  leather 
The  Carthaginians  had  a  kind  of  leather  money ;  and  the  Em- 
peror Frederick  Barbarossa,  1158,  and  John  the  Good,  King 
of  France,  1360,  also  issued  leather  money.  In  15*74,  when 
the  city  of  Leyden  was  besieged  by  the  Spaniards,  leather  money 
was  used,  and  even  quantities  of  pasteboard  were  coined  in  some 
parts  of  Holland.  In  the  13th  century,  money  made  out 
of  the  middle  bark  of  the  mulberry  tree,  cut  into  round  pieces 
and  stamped  with  the  mark  of  the  sovereign,  was  used  in 
China.  Cowry  shells  are  used  in  Africa,  in  India,  and  the  In- 
dian islands,  in  the  place  of  small  coins.  In  India  cakes  of  tea, 
in  China  pieces  of  silk,  in  Abyssinia  salt,  and  in  Iceland  and 
Newfoundland  codfish,  pass  for  money.  Wampum  was  used 
by  the  Indians ;  and  about  1635  was  the  prevailing  currency 
among  the  people  of  Massachusetts,  became  a  legal  tender,  and 
was  even  counterfeited.  About  the  same  time  corn  and  beans 
were  used,  and  musket  balls  passed  for  change  and  were  a  legal 
34 


530  THE   PHILOSOPHY    OF    ARITHMETIC. 

tender  for  sums  under  one  shilling.  Notched  wood  was  once 
used  as  a  currency  in  England.  So  late  as  1776,  it  was  custom- 
ary for  workmen  in  Scotland  to  carry  nails  as  money  to  the  bake- 
shop  and  ale-house.  It  is  thus  seen  that  barter  being  one  of  the 
prime  necessities  of  society,  man  finds,  amid  a  variety  of  things, 
some  one  or  more,  according  to  circumstances,  which  will  serve 
as  the  instrument  of  exchange. 

The  tendency,  however,  among  all  peoples,  has  been  towards 
the  precious  metals,  gold  and  silver.  These  metals,  though 
first  used  by  weight,  were  eventually  coined  into  pieces  of  con- 
venient and  fixed  values.  The  invention  of  coinage  has  been 
attributed  to  the  wife  of  Midas,  though  without  any  historical 
certainty.  High  authority  regards  the  Lydians,  about  1200  B. 
C,  as  the  inventors;  and  in  support  of  this  opinion  it  is  also 
claimed  that  the  earliest  electrum  coins,  which  undoubtedly  be- 
long to  cities  then  under  the  dominion  of  the  Lydian  kings, 
seem  to  be  of  greater  antiquity  than  any  in  the  entire  Greek 
series.  By  some  Greek  writers  the  invention  is  attributed  to 
Phidon,  King  of  Argos,  in  the  8th  century  B.  C. ;  but  he  is  now 
believed  to  have  merely  introduced  coinage  into  Greece.  It  is 
said  that  the  native  bronze  coins  of  China,  the  tsien  or  cash, 
bearing  the  inscription  tung-pan,  meaning  current  money,  had 
its  origin  about  1200  B.  C,  at  the  beginning  of  the  Chan  dy- 
nasty. Lycurgus  banished  gold  and  silver  and  made  the  money 
of  Sparta  of  iron,  $100  worth  of  which  required  a  cart  and  two 
oxen  to  draw  it. 

In  Rome,  for  nearly  500  years  after  its  foundation,  no  metal 
was  coined  but  copper  or  brass.  The  ses,  as,  or  libra,  was  a 
pound  weight  of  copper  or  brass,  stamped  by  the  state,  in  the 
reign  of  Servius  Tullius  (578-534  B.  C).  This  coin,  the  unit 
of  Roman  money,  was  originally  oblong  like  a  brick,  but  sub- 
sequently was  made  round;  and  was  cast,  not  struck.  Before 
this  reign,  unstamped  bars  of  copper  were  used  for  money 
Silver  was  first  coined  in  Rome  in  269  B.  C,  the  principal  coin 
being  the  denarius;  and  gold  in  207  B.  C,  although  it  is  held 


MEASURES   OF   VALUE.  531 

that  the  latter  did  not  form  a  part  of  the  regular  currency  until 
the  time  of  Julius  Caesar.  The  Emperors  possessed  the  privU 
lege  of  coining  gold  and  silver,  but  copper  could  be  coined  only 
by  decree  of  the  Senate. 

The  ancient  Britons,  at  the  invasion  of  Caesar,  had  money 
of  brass  and  iron,  and  it  was  paid  by  weight.  During  the 
reign  of  Augustus,  one  of  the  native  kings  caused  money  of 
gold,  silver,  and  brass  to  be  coined.  Under  the  Emperor 
Claudius,  the  Roman  money  took  the  place  of  the  Celtic,  and 
continued  in  circulation  until  after  the  withdrawal  of  the  Romans 
in  the  5th  century.  The  earliest  coins  subsequently  issued  are 
supposed  to  be  the  pennies  of  Ethelbert,  King  of  Kent  (560- 
616).  These  were  coarsely  stamped  with  the  king's  image  on 
one  side,  and  the  name  of  either  the  mint-master  or  the  city  in 
which  they  were  coined,  on  the  other  side.  At  this  time,  all 
money  accounts  began  to  be  expressed  in  pounds,  shillings, 
pence,  and  mancas  or  mancuses,  although  there  was  no  coin  but 
the  penny,  the  other  denominations  being  only  moneys  of  ac- 
count; 30  pence  made  a  manca,  5  pence  a  shilling,  and  40  shil- 
lings a  pound.  The  mancas  were  reckoned  both  in  gold  and 
silver.  In  King  Canute's  laws  the  distinction  is  made  that  a. 
mancuse  was  as  much  as  a  mark  of  silver,  while  a  manca  was 
a  square  piece  of  gold  valued  at  30  pence.  King  Athelstan 
(930)  decreed  that  money  should  be  uniform,  and  coined  only 
in  towns;  and  this  decree  mentions  the  fact  that  the  clergy 
shared  with  the  king  the  privilege  of  coining. 

The  Norman  kings  continued  to  coin  only  pence,  which  were 
of  silver,  and  with  a  cross  so  deeply  impressed  that  they  might 
easily  be  broken  into  halfpence  and  farthings.  The  word  ster- 
ling, to  denote  the  standard  money  of  England,  is  known  to 
have  been  used  as  early  as  during  the  reign  of  William  the 
Conqueror.  Severe  penalties  were  attached  to  the  counterfeit- 
ing of  money  by  Henry  I.  (1108),  and  during  his  reign  half- 
pence were  first  coined.  Henry  II.  (1154)  found  the  money  so 
much  debased  and  reduced  in  value,  that  he  provided  for  a  new 


532  THE    PHILOSOPHY    OF    ARITHMETIC. 

coinage,  and  punished  those  convicted  of  tampering  with  it. 
Silver  farthings  were  first  coined  in  1222.  In  1248  it  was  found 
that  the  money  of  the  realm  had  been  so  clipped  and  otherwise 
defaced  that  its  real  worth  bore  no  fixed  proportion  to  its  nom- 
inal value.  Henry  III.  therefore  ordered  that  the  old  coins 
should  be  brought  to  the  mint  and  exchanged  for  new  ones, 
weight  for  weight,  thus  entailing  the  entire  loss,  which  was 
very  great,  upon  the  actual  holders  of  these  coins,  which  justly 
caused  great  complaint.  During  this  reign,  in  1257,  gold  pen- 
nies were  first  coined,  which  weighed  y^-  of  a  pound  tower, 
and  passed  for  twenty  pence.  In  1279,  Edward  I.  caused  a 
new  coinage  of  halfpence  and  farthings  to  be  made,  and  pro- 
vided that  the  old,  which  were  principally  mere  fractions  cut 
to  suit,  should  no  longer  pass  current.  In  1300  he  positively 
prohibited  the  circulation  of  any  money  not  of  his  own  coinage. 
In  1301  he  diminished  the  weight  of  the  pound  sterling  three 
pennies,  equal  to  one  per  cent.  Edward  III.  (1335),  having 
exhausted  his  exchequer  and  embarrassed  himself  in  his  efforts 
to  conquer  France,  ordered  (1344)  that  in  future  266  pennies 
should  be  made  from  a  pound  sterling;  and  two  years  subse- 
quently he  increased  the  number  to  270  pennies.  Shilling 
pieces  were  first  coined  during  the  reign  of  Henry  VII.,  in 
1505 ;  and  in  1523,  during  the  reign  of  Henry  VIII.,  silver 
farthings  were  coined  for  the  last  time.  Queen  Elizabeth  raised 
the  standard  of  silver  coin,  and  in  1601  coined  for  Ireland  shil- 
lings, sixpences,  and  threepences  of  a  baser  kind.  The  circu- 
lation of  leaden  tokens  issued  by  the  tradesmen  of  London 
was,  to  a  great  extent,  stopped  about  the  beginning  of  the  17th 
century.  James  I.,  in  1613,  debased  a  portion  of  the  coin, 
having  coins  in  circulation  of  two  qualities  of  fineness.  James 
II.  (1685-8)  issued  coins  of  tin,  and  authorized  those  of  gun 
metal  and  of  pewter.  The  first  sovereigns  were  coined  in  1489, 
under  Henry  VII.;  half,  quarter,  and  eighth  sovereigns  by 
Henry  VIII.,  in  1544;  and  the  first  guinea  by  Charles  II.,  in 
1675. 


MEASURES    OE   VALUE.  533 

Up  to  the  18th  century,  England  had  a  double  monetary 
standard,  gold  and  silver;  but  owing  to  the  over-valuation  of 
silver  in  France,  heavy  silver  coins  disappeared  from  circula- 
tion, and  the  evil  became  so  great  that  in  1774  it  was  declared 
that  silver  should  no  longer  be  a  legal  tender,  except  by  weight, 
beyond  £25.  In  1816  the  pound  standard  of  silver  was  coined 
into  66s.,  the  relative  value  with  gold  being  as  1  to  14.287. 
Silver  then  became  a  legal  tender  for  only  40  shillings.  In 
1792  Congress  fixed  the  relative  value  of  silver  and  gold  at  1 
to  15,  which  over-valuation  of  silver  caused  gold  to  be  exported 
in  such  quantities  that  it  was  impossible  to  maintain  a  gold 
circulation.  In  1834  the  standard  was  changed  to  1  to  16, 
while  with  other  nations  it  was  1  to  15^,  which  caused  silver 
to  be  so  largely  exported  that  in  1853  the  ratio  was  changed 
to  1  to  14.88,  and  silver  was  made  a  legal  tender  only  for  sums 
under  $5.  By  the  coinage  act  of  1873,  it  was  again  changed 
to  1  to  14.95.  For  other  interesting  information  upon  this 
subject,  see  the  American  Cyclopedia,  from  which  most  of 
these  facts  have  been  taken. 

Money. — That  by  which  value  is  estimated  is  called  Money 
Money  may  be  defined  as  the  measure  or  representative  of  the 
value  of  things.  It  is  so  called  from  the  temple  of  Juno 
Moneti,  in  which  money  was  first  coined  at  Rome.  It  is  of 
two  kinds — coin  and  paper  money.  The  money  of  a  country 
is  called  its  currency,  from  curro,  I  run,  on  account  of  its 
circulating  through  the  country.  The  coin  of  a  country  is 
called  its  specie  currency,  and  the  paper  money  its  paper 
currency.  Coin  is  metal  prepared  to  circulate  as  money. 
The  metals  used  in  this  country  are  gold,  silver,  copper,  and 
nickel.  Paper  money  consists  of  printed  promises  to  pay  the 
bearer  a  certain  amount,  duly  authorized  to  circulate  as  money 

United  States  Money. — The  present  system  of  our  currency 
was  established  by  an  Act  of  Congress,  August  8th,  1786. 
It  was  formerly,  and  is  still  sometimes,  called  Federal  Money, 
because  it  was  the  money  of  the  Federal  Union.     A  plan  for 


534  THE    PHILOSOPHY    OF    ARITHMETIC. 

an  American  coinage  was  submitted  to  Congress  in  1782,  by 
Robert  Morris,  the  head  of  the  Finance  Department,  though 
its  authorship  is  claimed  for  Gouverneur  Morris.  The  plan 
adopted  was  that  presented  by  Thomas  Jefferson. 

The  standard  unit  of  the  system  is  the  dollar,  and  the  scale 
was  made  to  conform  to  the  decimal  system  of  notation.  The 
term  dollar  is  probably  of  German  origin,  derived  from  thai, 
signifying  a  dale  or  valley.  It  is  supposed  that  they  were 
first  coined  about  the  year  1518,  at  Joachimsthal  (Joachim's- 
valley),  a  mining  town  of  Bohemia,  and  called  Joachims-thaler, 
and  finally  abbreviated  to  thaler.  There  are,  however,  several 
other  theories  to  account  for  the  word.  Some  German  scholar 
derive  the  word  thaler  from  talent,  which  was  used,  in  the 
Middle  Ages,  to  denote  a  pound  of  gold.  Tooke  says  it  is 
from  the  Anglo-Saxon  dael,  a  portion,  being  a  part  or  portion 
of  a  ducat.  Thompson  thinks  that  it  is  from  the  Swedish 
daler,  from  the  town  Dale  or  Daleberg,  where  it  was  coined. 
The  dollar  is  a  silver  coin  of  Germany,  Holland,  Spain,  Mexico, 
etc.,  though  its  value  is  not  the  same  in  all  countries.  In 
Spain,  the  coin  is  called  dalera,  the  famous  Spanish  dollar, 
which  for  centuries  figured  so  conspicuously  in  the  commerce 
of  the  world.  The  Spanish  dollar,  called  also  the  milled  dol- 
lar, from  its  milled  edge,  was  taken  as  the  basis  of  United 
States  coin  and  money  account. 

The  term  dime,  one-tenth  of  a  dollar,  is  derived  from  the 
French  disme,  meaning  ten  ;  the  term  cent,  one  hundredth  of 
a  dollar,  from  the  Latin  centum,  a  hundred ;  the  term  mill,  one 
thousandth  of  a  dollar,  from  the  Latin  mille,  a  thousand.  The 
term  eagle  is  probably  applied  on  account  of  the  design  on  the 
coin.  The  cent  was  proposed  in  1782,  by  Robert  Morris,  and 
was  named  by  Thomas  Jefferson  three  years  later.  It  was  first 
coined  in  Vermont,  in  1785,  in  the  town  of  Rupert,  and  in  the- 
same  year  by  Connecticut  at  New  Haven,  and  by  New  Jersey 
and  Massachusetts  in  1786.  The  same  year  Congress  author- 
ized the  establishment  of  a  mint,  but  in  1787  they  contracted 


MEASURES   OF   VALUE.  535 

with  James  Jarvis  for  300  tons  of  cents,  which  were  coined  in 
New  Haven.  In  1792  a  national  mint  was  finally  established 
under  regulations  which  continued  in  force  over  forty  years. 
The  cent  bore  the  head  of  Washington  on  one  side  and  a  chain 
of  thirteen  links  on  the  other.  The  French  Revolution  creat- 
ing a  rage  in  America  for  French  ideas,  the  image  of  Wash- 
ington was  deposed  and  the  head  of  the  Goddess  of  Liberty 
took  its  place,  the  chain  being  also  replaced  by  the  olive  wreath 
of  peace.  French  liberty  was  short-lived  and  so  was  the  image 
upon  the  coin.  The  present  face,  with  its  classic  features,  was 
subsequently  adopted,  and  has  been  but  slightly  changed  since 
its  adoption. 

The  origin  of  the  symbol  $,  has  never  been  satisfactorily 
determined.  There  are  several  theories,  the  most  important  of 
which  are  the  following:  1st.  It  is  supposed  to  be  a  combina- 
tion of  U.  S.,  signifying  United  States,  formed  by  writing  the 
U  over  the  S,  which  became  changed  in  course  of  time  to  its 
present  form.  2d.  It  is  said  to  be  a  modification  of  the  figure 
8,  denoting  a  piece  of  eight  Reals  or  Testons.  The  dollar  was 
formerly  called  "a  piece  of  eight,"  and  designated  by  the  symbol 
■£.  3d.  It  is  said  to  be  derived  from  the  representation  of  the 
two  "  Pillars  of  Hercules,"  the  ancient  name  of  the  opposite 
promontories  at  the  Straits  of  Gibraltar.  These  were  repre- 
sented by  two  vertical  lines  connected  with  a  scroll  or  label, 
and  the  coins  containing  this  mark  were  called  pillar  dollars 
4th.  It  is  said  to  be  a  combination  of  HS.,  the  mark  of  the 
Roman  money  unit.  This  symbol  was  prefixed  to  the  numer- 
als representing  any  sum,  as  the  dollar  mark  is  employed  by 
us.  The  symbol  HS.  is  a  contraction  of  II.,  two  and  Semis,  half, 
meaning  two  and  a  half;  being  equivalent  to  the  word  Sester- 
tius, which  was  equal  to  two  and  one-half  nummi.  The  Ses- 
terce was  the  Roman  money  unit,  as  the  dollar  is  ours.  5th. 
It  is  said  to  be  a  combination  of  P  and  S.,  from  peso  duro,  or 
peso  fuerte,  meaning  "hard  dollar."  In  Spanish  accounts 
this  is  always  abbreviated  by  writing  an  S  over  a  P,  and  plac- 


536  THE    PHILOSOPHY    OF    ARITHMETIC. 

ing  the  sign  after  the  sum,  as  is  also  customary  among  the 
Portuguese. 

The  coins  are  of  gold,  silver,  bronze,  and  nickel  The  gold 
coins  are  the  double  eagle,  eagle,  half -eagle,  quarter-eagle,  three 
dollar,  and  one  dollar.  Fifty-dollar,  half-dollar,  and  quarter- 
dollar  pieces  are  also  coined,  but  are  not  legal  circulation.  The 
silver  coins  are  the  dollar,  half-dollar,  quarter-dollar,  dime, 
half-dime,  and  three-cent  piece.  The  bronze  coins  are  the  two- 
cent  piece  and  the  cent.  The  half-cent  and  cent  of  pure  copper 
are  not  now  coined.  The  mill  has  never  been  a  coin,  it  is 
merely  a  convenient  name  for  the  tenth  part  of  a  cent.  The 
nickel  coins  are  the  Jive-cent  and  three-cent  pieces. 

The  gold  and  silver  coins  consist  of  nine  parts  pure  metal 
and  one  part  alloy,  except  the  three-cent  piece,  which  is  one- 
fourth  alloy.  The  alloy  of  the  silver  coin  is  pure  copper ;  that 
of  the  gold  coin  is  copper,  or  copper  and  silver,  the  silver  not 
to  exceed  one-tenth  of  the  whole  alloy.  The  nickel  coins  con- 
tain one  part  nickel  and  three  parts  copper.  The  bronze  coins 
contain  95  per  cent,  of  copper  and  5  per  cent,  of  tin  and  zinc. 
The  eagle  weighs  258  grains,  the  other  gold  coins  in  proportion ; 
the  silver  trade  dollar,  intended  for  commerce  with  China  and 
Japan,  weighs  420  grains  ;  the  half-dollar,  192.9  grains,  and  the 
other  silver  coins  in  proportion ;  the  nickel  five-cent  piece  weighs 
5  grams  (IT.  16  grains  nearly),  and  the  three-cent  piece  weighs 
30  grains ;  the  bronze  cent  weighs  48  grains.  The  half-dollar, 
being  half  the  weight  of  the  five-franc  piece  of  France,  Bel- 
gium, and  Switzerland,  of  the  five-lire  piece  of  Italy,  the  five- 
peseta  coin  of  Spain,  the  five-drachma  coin  of  Greece,  and  being 
equal  in  weight  to  the  silver  florin  of  Austria,  is  a  step  towards 
an  international  coinage. 

Previous  to  the  establishment  of  the  decimal  currency,  we 
employed  the  currency  of  England,  that  is,  pounds,  shillings, 
and  pence.  Several  of  the  States  still  use  shillings  and  pence, 
though  not  with  the  same  values.  The  difference  in  the  num- 
ber of  shillings  required  for  a  dollar  in  the  different  States  is 


MEASURES    OF    VALUE.  537 

owing  to  a  depreciation  in  the  paper  money  issued  by  the 
colonies.  This  depreciation  was  so  great  that  in  1749,  £1100 
currency  was  only  equal  to  £100  sterling.  Soon  after  this 
Massachusetts  receiving  a  remittance  from  England,  called  in 
her  depreciated  money  at  the  rate  of  a  Spanish  dollar  to  45 
shillings  of  the  paper  currency,  and  the  Legislature  also  passed 
an  act  fixing  the  par  exchange  between  Massachusetts  and  Eng- 
land at  £133^  currency  to  £100  sterling,  and  6  shillings  to  the 
Spanish  dollar.  A  similar  depreciation  of  the  paper  money  es- 
tablished the  currencies  of  the  other  colonies.  From  this  diver- 
sity in  the  colonial  currencies  it  happens  that  the  Spanish  real 
of  one-eighth  of  a  dollar,  was  called  in  New  England  ninepence  ; 
in  New  York,  one  shilling ;  in  Pennsylvania,  elevenpence  or 
a  levy. 

The  earliest  coinage  in  America  was  made  in  1612,  at  the 
Somers  Islands,  now  called  Bermudas.  The  coin  was  of  brass, 
with  the  legend  "Sommer  Island,"  and  "a  hogge  on  one  side, 
in  memory  of  the  abundance  of  hogges  which  were  found  on 
their  first  landing."  In  1645,  the  assembly  of  Virginia  pro- 
vided for  a  copper  coinage,  but  the  law  was  never  executed 
The  earliest  colonial  coinage  was  in  Massachusetts,  under  an 
act  passed  May  27,  1652,  which  established  a  "mint  howse"  at 
Boston.  The  first  coins  were  found  to  be  too  plain  to  prevent 
"  washing  and  clipping,"  and  were  afterwards  stamped  with  a 
figure  of  a  tree,  whence  they  were  called  "pine-tree  shillings." 
In  1662  the  assembly  of  Maryland  passed  an  act  "for  the 
setting  up  of  a  mint  within  the  province  ;"  but  it  seems  never 
to  have  been  established.  George  I.  attempted  to  introduce 
into  the  colonies,  coins  made  of  Bath  metal,  or  pinchbeck ;  but 
this  money  was  rejected  by  them.  From  1778  to  1787,  the 
power  of  coinage  was  exercised  both  by  the  confederation  in 
Congress  and  by  several  of  the  individual  states.  The  mint 
established  in  1792  continued  in  operation  under  nearly  the 
same  regulations  up  to  1837,  since  which  time  numerous 
changes  have  been  made,  in  both  the  value  and  the  composi- 
23* 


538  THE    PHILOSOPHY    OF    ARITHMETIC. 

tion  of  the  coins.  In  the  United  States,  the  right  of  coinage 
is  vested  by  the  Constitution  in  Congress,  and  prohibited  to 
the  several  states ;  and  yet  individuals  are  left  free  to  coin 
money,  provided  that  the  coins  be  not  in  "resemblance  or 
similitude"  of  the  gold  or  silver  coins  issued  from  the  mint. 
Large  amounts  of  private  gold  coins  have  been  struck  and 
circulated  in  different  parts  of  the  country.  In  the  case  of 
copper  coins,  however,  the  offering  or  receiving  of  any  other 
copper  coins  than  the  cent  and  half-cent  is  prohibited  by  fine 
and  forfeiture. 

English  Money. — English,  or  Sterling  Money,  is  the  legal 
currency  of  England.  The  scale  is  irregular,  ascending  by  4, 
12,  20.  The  term  Sterling  is  supposed  to  be  derived  from 
Easterling,  the  popular  name  of  the  Baltic  and  German  traders 
who  visited  London  in  the  Middle  Ages.  In  what  manner  it 
came  to  be  so  applied  is  not  certainly  known.  Camden  says 
from  the  employment  of  German  artists  in  coining.  It  is  gen- 
erally supposed  that,these  traders  being  called  Easterlings,  their 
money  would  naturally  be  called  Easterling  money,  which  was 
finally  changed,  by  use,  to  Sterling  money.  The  silver  penny 
was  first  called  Easterling.  The  unit  is  the  pouud,  represented 
by  the  sovereign,  and  the  £1  bank  note. 

The  term  pound,  as  a  measure  of  value,  is  derived  from  pound, 
as  a  measure  of  weight,  from  the  fact  that  anciently  240  pence 
were  equal  to  a  pound  in  weight,  the  term  originally  signifying 
a  weight  and  not  a  value  of  money.  The  penny  was  formerly 
a  silver  piece,  first  coined  by  the  Saxons.  The  term  farthing 
is  from  four  things.  Previous  to  the  time  of  Edward  I.,  the 
penny  was  struck  with  a  cross  so  deeply  sunk  in  it,  that  it  could 
be  easily  broken  into  halves  and  quarters,  whence  the  names 
half  penny  and  four  things  or  farthings.  Edward  I.  reduced 
the  penny  to  a  fixed  standard,  fixing  its  weight  at  the  thirtieth 
part  of  an  ounce.  It  afterward  suffered  successive  diminutions 
until  the  reign  of  Elizabeth,  when  its  value  was  fixed  at  the 
sixty-second  part  of  an  ounce  of  silver,  which  standard  is  still 


MEASURES   OF   VALUE.  539 

observed.  The  shilling,  among  the  ancient  Saxons,  was  only 
five  pence.  It  subsequently  underwent  many  alterations,  con- 
taining sometimes  16  pence,  and  sometimes  20  pence.  Its  pre- 
sent value  was  fixed  during  the  reign  of  Edward  I.  A  coin 
of  the  same  name  is  found  in  several  other  countries.  The 
symbols  £.,  s.,  d.,  qr.,  are  the  initials  of  the  Latin  words  libra, 
solidus,  denarius,  and  quadrans ;  signifying  respectively  pound, 
shilling,  penny,  and  quarter.  The  old  /,  the  original  abbrevia- 
tion for  shillings,  was  formerly  written  between  shillings  and 
pence  ;  thus  7  s.  6d.  was  written  7/6.  The  /  has  since  been 
changed  into  /,  and  shillings  and  pence  are  sometimes  written 
thus,  7/6. 

The  English  gold  coins  are  the  5  sovereign  piece,  the  double- 
sovereign,  the  sovereign  and  half-sovereign,  the  guinea  and 
half-guinea.  The  Sovereign,  equal  to  20  shillings,  represents 
the  pound  sterling.  Its  legal  value  in  our  currency  is  $4.8665. 
It  is  the  standard  gold  coin.  The  Guinea,  equal  to  21  shillings, 
was  first  coined  during  the  reign  of  Charles  II.,  in  the  year 
1662,  of  gold  brought  from  Guinea,  and  hence  its  name.  The 
guinea  and  half-guinea  are  no  longer  coined,  though  some  of 
them  are  still  in  circulation.  The  silver  coins  are  the  crown, 
the  half-crown,  the  florin,  the  shilling,  the  sixpenny,  four- 
penny,  and  threepenny  piece.  The  Crown  is  an  old  English 
coin,  stamped  with  the  figure  of  a  crown,  whence  its  name. 
Its  value  is  5  shillings  sterling.  The  copper  coins  are  the 
penny,  half -penny,  and  farthing.  The  groat,  worth  4  pence,  is 
often  mentioned. 

The  noble,  the  angel,  and  the  mark,  are  old  gold  coins  no 
longer  in  use.  The  Noble  is  an  old  coin  of  the  Middle  Ages, 
coined  in  the  reign  of  Edward  III.  Its  value  is  6  shillings 
8  pence.  The  Angel  is  an  old  coin  valued  at  10  shillings.  It 
was  impressed  with  the  figure  of  an  angel,  in  commemoration 
of  a  saying  of  Pope  Gregory  I.,  that  the  pagan  Angli,  or  Eng- 
lish, were  so  beautiful,  that  were  they  Christians,  they  would 
be  angels.     The  Mark  is  an  old  coin,  current  in  England  and 


540  THE    PHILOSOPHY    OF    ARITHMETIC. 

Scotland,  valued  at  13  shillings  4  pence.  A  piece  of  money 
bearing  this  name,  valued  at  1  shilling  4  pence,  is  at  present 
used  in  Hamburg. 

The  standard  for  gold  coins  is  22  carats  fine,  that  is,  11  parts 
pure  gold  and  one  part  alloy.  This  makes  the  English  standard 
jL  alloy,  while  the  standard  of  the  United  States  is  ^  alloy. 
The  standard  for  silver  is  37  parts  pure  silver  and  3  parts  alloy, 
hence  the  silver  coins  are  f£  pure  and  ^  copper.  Pence  and 
half-pence  are  made  of  pure  copper.  The  sovereign  weighs 
123.274  grains;  the  shilling  weighs  87.27  grains;  the  penny 
weighs  240  grains,  or  ^  ounce  Troy. 

The  currency  of  Canada  is  the  same  as  that  of  the  United 
States,  the  table  and  denominations  also  being  the  same.  The 
decimal  currency  was  adopted  in  1858,  the  Act  taking  effect  in 
1859,  previous  to  which  their  currency  was  the  same  as  the 
English.  The  coins  consist  of  silver  and  copper.  The  silver 
coins  are  the  fifty-cent  piece,  the  twenty-five-cent  piece,  the 
shilling  or  twenty-cent  piece,  the  dime,  and  the  half-dime. 
The  copper  coin  is  the  cent.  The  shilling  equals  about  19 
cents  of  United  States  money;  the  values  of  the  other  silver 
coins  are  proportional.  The  silver  coins  consist  of  37  parts 
silver  to  3  of  copper,  the  same  as  the  English  silver  coins. 
There  is  no  gold  coinage,  the  British  and  American  gold  coins 
being  a  legal  tender. 

French  Money. — The  French  system,  like  our  own,  is  founded 
upon  the  decimal  notation.  The  unit  is  the  franc,  whose 
value  is  fixed,  by  a  late  Act  of  Congress,  at  19.3  cents.  The 
franc  is  divided  into  tenths  and  hundredths,  called  respectively 
decimes  and  centimes.  The  decime,  like  our  dime,  is  not  used 
in  business  calculations,  but  is  expressed  by  centimes.  The 
gold  and  silver  coins  are  nine-tenths  pure  metal. 

German  Money. — A  new  and  uniform  system  of  coinage  has 
been  adopted  by  the  New  German  Empire.  The  unit  is  the 
reichsmark,  worth  23.85  cents.  A  pound  of  gold,  .9  pure,  is 
divided  into  139  J  coins,  and  the  tenth  part  of  this  coin  is  called 
a  mark,  and  this  is  subdivided  into  100  pfennige. 


CHAPTER  Y. 

MEASURES   OF   TIME. 

TIME,  in  the  sense  here  used,  is  regarded  as  a  limited 
portion  of  duration,  measured  by  certain  conventional  or 
natural  periods.  It  is  a  definite  portion  of  absolute  Time, 
which  is  duration  without  beginning  or  end.  The  idea  of  Time 
in  the  absolute,  is  a  grand  intuition,  like  Space  ;  Time,  here 
considered,  is  known  by  experience  and  judgment. 

Measures  of  Time  were  also  originally  derived  from  nature. 
The  simplest  unit  of  time,  the  day,  "  nature  supplies  ready 
made."  The  next  simplest  period,  the  month,  or  mooneth,  is 
also  presented  to  us  by  the  changes  of  the  moon  constituting  a 
lunation.  For  larger  divisions  than  these,  the  phenomena  of 
the  seasons  and  the  chief  events  from  time  to  time  occurring, 
have  been  used  by  early  and  uncivilized  races.  Among  the 
Egyptians  the  rising  of  the  Nile  served  as  a  point  from  which 
to  reckon  time.  The  New  Zealanders  were  found  to  begin  their 
year  from  the  rising  of  the  Pleiades  above  the  sea.  The 
migration  of  birds  indicated  the  season  to  the  Greeks  and  other 
nations.  The  Hottentot  denoted  periods  by  the  number  of 
moons  before  or  after  the  ripening  of  his  chief  article  of  food. 
Barrow  states  that  the  Kaffir  chronology  is  kept  by  the  moon, 
and  is  registered  by  notches  on  sticks — the  death  of  a  favorite 
chief,  or  the  gaining  of  a  victory,  serving  for  a  new  era.  The 
peasantry  of  England  refer  to  occurrences  as  "before  sheep- 
shearing;"  and  in  this  country  we  often  hear  "harvest  time," 
•'  after  harvest,"  etc.,  used  as  dates  of  reckoning.  It  is,  there- 
fore, manifest  that  the  more  or  less  equal  periods  perceived  in 

(541) 


542  THE    PHILuovtHY    OF    ARITHMETIC. 

Nature  gave  the  first  units  of  the  measure  of  time,  as  i~  nK^ 
cases  already  considered. 

Historical. — As  society  progressed  in  civilization,  the  neces- 
sity of  smaller  and  more  precise  units  became  apparent,  whic: 
at  last  led  to  the  adoption  of  some  regular  system  adapted  to 
the  purposes  of  civil  life.  Such  a  system  is  called  a  Calendar, 
from  the  Latin  calare,  to  call.  In  the  early  days  of  Rome  it 
was  the  custom  of  the  Pontiff  to  call  the  people  together  on 
the  first  day  of  each  month,  to  apprise  them  of  the  days  that 
were  to  be  kept  sacred  during  the  month.  Hence  dies  calendee, 
the  calends  or  first  days  of  the  different  months. 

The  present  calendars  are  derived  from  the  Romans.  Rom- 
ulus is  supposed  to  have  first  undertaken  to  divide  the  year  in 
such  a  manner,  that  certain  epochs  should  return  periodically 
after  the  revolution  of  the  sun ;  but  the  knowledge  of  astron- 
omy was  not  then  sufficiently  advanced  to  allow  this  to  be  done 
with  much  precision.  He  placed  the  beginning  of  the  year  in 
the  spring,  and  divided  it  into  ten  months, — March,  April, 
May,  June,  Quintilis,  Sextilis,  September,  October,  November, 
and  December.  March,  May,  Quintilis,  and  October,  contained 
31  days  each  ;  the  other  six  contained  only  thirty.  The  names 
Quintilis  and  Sextilis  remained  in  the  calendar  till  the  end  of 
the  republic,  when  they  were  changed  into  July  and  August ; 
the  former  in  honor  of  Julius  Caesar,  and  the  latter  of  Augustus. 
The  Roman  month  was  divided  into  three  periods  by  the 
Calends,  the  Nones,  and  the  Ides.  The  Calends  were  invaria- 
bly placed  at  the  beginning  of  the  month  ;  the  Ides,  at  the  mid- 
dle of  the  month,  on  the  13th  or  15th;  and  the  Nones  (novem, 
nine)  were  the  ninth  day  before  the  Ides,  counting  inclusively. 
From  these  three  terms  the  days  were  counted  backward  in  the 
following  manner:  those  days  comprised  between  the  calends 
and  the  nones  were  denominated  days  before  the  nones;  those 
between  the  nones  and  the  ides,  days  before  the  ides;  and 
those  from  the  ides  to  the  end  of  the  month,  days  before  the 
calends.     Hence  the  phrases  pridie  calendas,  tertio  calendar 


MEASURES   OF    TIME.  543 

etc.,  meaning  the  second  day  before  the  calends,  or  last  day  of 
the  month,  the  third  day  before  the  calends,  or  last  but  one  of 
the  month  (the  calends  being  included  in  the  reckoning),  and  so 
on.  In  the  months  of  March,  May,  July,  and  October,  the 
ides  fell  on  the  15th,  and  the  nones,  consequently,  on  the  7th. 
In  all  other  months  the  ides  fell  on  the  13th,  and  the  nones, 
consequently,  on  the  5th.  The  number  of  days  receiving  their 
denomination  from  the  calends,  depended  on  the  number  of 
days  in  the  month,  and  the  day  on  which  the  ides  fell.  For 
example,  if  the  month  had  thirty-one  days  and  the  ides  fell  on 
the  13th  (as  in  January,  August,  and  December),  there  would 
remain  eighteen  days  after  the  ides,  which,  added  to  the  first 
of  the  following  month,  made  nineteen  days  of  calends.  Hence 
January  14th  was  styled  the  nineteenth  day  before  the  calends 
of  February,  and  so  on. 

The  year  of  Romulus,  according  to  the  mythical  history, 
contained  only  304  days.  Numa,  it  is  said,  added  two  months ; 
January  to  the  beginning  of  the  year,  and  February  to  the 
end.  About  the  year  452  B.  C.  this  arrangement  was  changed 
by  the  Decemvirs,  who  placed  February  after  January ;  since 
that  time  the  order  of  the  months  has  remained  undisturbed. 
In  Numa's  year  the  months  consisted  of  29  and  30  days  alter- 
nately, to  correspond  with  the  synodic  revolution  of  the  moon. 
The  year  would  therefore  consist  of  354  days;  but  one  day 
^as  added  to  make  the  number  odd,  as  being  more  lucky.  In 
order  to  produce  a  correspondence  with  the  solar  year,  Numa 
ordered  an  intercalary  month  to  be  inserted  every  second  year 
between  the  23d  and  24th  of  February,  consisting  alternately  of 
22  and  23  days.  Had  this  regulation  been  strictly  adhered  to, 
the  mean  length  of  the  year  would  have  been  365J  days,  and 
the  months  would  have  continued  for  a  long  time  to  correspond 
with  the  same  seasons.  But  a  discretionary  power  over  the 
intercalary  month  was  exercised  by  the  pontiffs,  for  the  purpose 
of  hastening  or  retarding  the  days  of  election  of  magistrates ; 
and  thus  the  Roman  calendar  continued  in   a  state  of  uncer- 


544  THE    PHILOSOPHY    OF    ARITHMETIC. 

tainty  aiid  confusion  till  the  time  of  Julius  Caesa.*,  when  the 
civil  equinox  differed  from  the  astronomical  by  three  months. 

Under  the  advice  of  the  astronomer  Sosigenes,  Caesar  abol- 
ished the  lunar  year,  and  regulated  the  civil-  year  entirely  by 
the  sun.  He  decreed  that  the  common  year  should  consist  of 
365  days;  but  that  every  fourth  year  should  contain  366.  In 
distributing  the  days  among  the  different  months,  he  ordered 
that  the  odd  months  should  contain  each  31  days  and  the  even 
months  30  ;  excepting  February,  which  in  common  years  was 
to  contain  only  29  days,  but  every  fourth  year  30  days.  This 
natural  and  convenient  arrangement  was  interrupted  to  gratify 
the  frivolous  vanity  of  Augustus,  by  giving  August,  the  month 
named  after  him,  an  equal  number  of  days  with  July,  which 
was  named  after  Julius  Caesar.  The  intercalary  day  which 
occurred  every  fourth  year  was  inserted  between  the  24th  and 
25th  of  February.  According  to  the  peculiar  and  awkward 
manner  of  reckoning  adopted  by  the  Romans,  the  24th  of  Feb- 
ruary was  called  the  6th  before  the  calends  of  March,  sexto  cal- 
endas.  In  the  intercalary  year  this  day  was  repeated,  and 
called  bis-sexto  calendas ;  whence  the  term  bissextile.  The  cor- 
responding English  term  leap  year,  appears  less  correct,  as  it 
seems  to  imply  that  a  day  was  leapt  over  instead  of  being 
thrust  in.  It  may  be  remarked  that  in  the  ecclesiastical  calen- 
dar, the  intercalary  day  is  still  inserted  between  the  24th  and 
25th  of  February. 

The  Julian  year  consisted  of  365^  days,  and  consequently 
differed  in  excess  by  11  minutes,  10.35  seconds  from  the  true 
solar  year,  which  consists  of  365  days,  5  hours,  48  minutes, 
49.7  seconds.  In  consequence  of  this  difference  the  astrono- 
mical equinox,  in  the  course  of  a  few  centuries,  sensibly  fell 
back  towards  the  beginning  of  the  year.  In  the  time  of  Julius 
Caesar  it  corresponded  with  the  25th  of  March;  in  the  16th 
century  it  had  retrograded  to  the  11th.  Th^  correction  of  this 
error  was  one  of  the  purposes  sought  to  be  obtained  by  the 
reformation  of  the  calendar  effected  by  Pope    Gregory  XIII 


MEASURES   OF   TIME.  545 

In  1582.  By  suppressing  10  days  in  the  calendar,  Gregory 
restored  the  equinox  to  the  21st  of  March,  the  day  on  which  it 
fell  at  the  time  of  the  Council  of  Nice,  in  325.  In  order  that 
the  same  inconvenience  might  be  prevented  in  the  future,  he 
ordered  the  intercalation,  which  took  place  every  fourth  year, 
to  be  omitted  in  years  ending  centuries;  excepting  on  the 
400th,  and  the  years  which  are  multiples  of  400.  By  this 
adjustment  of  the  calendar  the  difference  between  a  civil  and  a 
solar  year  will  amount  to  a  day  in  about  3,860  years. 

The  Gregorian  calendar  was  received  immediately  or  shortly 
after  its  promulgation  by  nearly  all  the  Roman  Catholic  coun- 
tries of  Europe.  The  Protestant  states  of  Germany  and  the 
kingdom  of  Denmark  adhered  to  the  Julian  calendar  until 
1700;  and  in  England  the  alteration  was  successfully  opposed 
by  popular  prejudices  until  1752.  In  that  year  the  Julian 
calendar,  or  old  style,  as  it  was  called,  was  formally  abolished 
by  the  Act  of  Parliament,  and  the  date  used  in  all  public 
transactions  rendered  coincident  with  that  followed  in  all 
European  countries,  by  enacting  that  the  day  following  the  2d 
of  September  of  the  year  1752  should  be  called  the  14th  of  that 
month.  When  the  alteration  was  made  by  Gregory  it  was 
only  necessary  to  drop  10  days;  the  year  1700  having  inter- 
vened, which  was  a  common  year  in  the  Gregorian  but  a  leap 
year  in  the  Julian  calendar,  it  was  now  necessary  to  drop  11 
days.  The  old  style  is  still  adhered  to  in  Russia  and  the 
countries  belonging  to  the  communion  of  the  Greek  Church; 
the  difference  of  date  in  the  present  century  amounts  to  12 
days. 

A  new  reform  of  the  calendar  was  attempted  in  France 
during  the  period  of  the  Revolution.  The  beginning  of  the 
year  was  fixed  at  the  autumnal  equinox,  which  nearly  coincided 
with  the  foundation  of  the  republic.  The  names  of  the  ancient 
months  were  abolished,  and  others  substituted  having  reference 
*,o  agricultural  labors,  or  the  state  of  nature  in  different  seasons 
of  the  year.  But  the  change  was  found  to  be  inconvenient  and 
impracticable,  and  after  a  few  vears  was  formally  abandoned, 
35 


546  THE    PHILOSOPHY    OF    ARITHMETIC. 

Beginning  of  the  Year. — There  has  been  great  diversity 
among  different  nations  in  the  time  of  beginning  the  year. 
The  ancient  Egyptians,  Chaldeans,  Persians,  Syrians,  Phoeni- 
cians, Carthaginians,  each  began  their  year  at  the  autumnal 
equinox,  about  the  22d  of  September.  The  Jews  began  their 
civil  year  at  the  same  time,  but  their  ecclesiastical  year  dated 
from  the  vernal  equinox,  about  March  22d.  Among  the 
Greeks,  until  432  B.  C  ,  when  Meton  introduced  the  cycle 
named  after  him,  the  year  commenced  at  the  winter  solstice, 
about  December  22d;  and  subsequently  at  the  summer  solstice, 
about  the  22d  of  June.  The  Roman  year  from  the  time  of 
Nunia  began  at  the  winter  solstice.  Among  the  Latin  Chris- 
tian nations  there  were  seven  different  dates  for  the  beginning 
of  the  year :  March  1 ;  January  1 ;  December  25 ;  March  25 
(beginning  the  year  more  than  nine  months  sooner  than  we 
do,  called  the  Pisan  calculation) ;  March  25  (beginning  nearly 
three  months  later  than  we  do,  called  the  Florentine  calcula- 
tion); at  Easter;  and  on  January  1  (one  year  in  advance  of 
us).  In  France  the  year  began  in  general  at  March  1,  under 
the  Merovingians;  at  December  25,  under  the  Carlovingians; 
and  at  Easter  under  the  Capetians.  By  edict  of  Charles  IX., 
in  1564,  the  beginning  of  the  year  was  ordered  at  January  1. 
In  England,  from  the  14th  century  till  the  change  of  style  in 
1752,  the  legal  and  ecclesiastical  year  began  at  March  25,  the 
day  of  the  Annunciation,  though  it  was  not  uncommon  to 
reckon  it  in  writing  from  January  1,  the  day  of  the  circumcision. 
After  the  change,  events  which  had  occurred  before  March  25 
in  the  old  legal  year,  would  by  the  new  arrangement,  be 
reckoned  in  the  next  subsequent  year.  Thus  the  revolution  of 
1688  occurred  in  February  of  that  legal  year,  or,  as  we  would 
now  say,  in  February,  1689;  and  it  was  at  one  time  customary 
to  write  the  date  thus, — February,  168f. 

The  ancient  northern  nations  of  Europe  began  their  year  from 
the  winter  solstice.  In  the  era  of  Constantinople,  which  was 
in  use  in  the  Byzantine  Empire,  and  in  Russia  till  the  time  of 


MEASURES   OF    TIME.  547 

Peter  the  Great,  the  civil  year  began  with  September  1,  and 
the  ecclesiastical  sometimes  with  March  21,  and  sometimes  with 
April  L  The  beginning  of  the  Mohammedan  year,  which  is 
iunar,  is  not  at  any  fixed  time,  but  retrogrades  through  the  dif- 
ferent seasons  of  the  solar  year.  The  later  Jewish  year  is 
lunar,  but  by  the  intercalation  of  a  thirteenth  month,  seven  times 
in  a  cycle  of  nineteen  years,  is  brought  into  harmony  with  the 
solar  periods ;  it  begins  at  the  autumnal  equinox.  Among  most 
of  the  peoples  of  the  East  Indies  the  year  is  lunar,  and  begins 
with  the  first  quarter  of  the  moon  the  nearest  to  the  beginning 
of  December.  Among  the  Peruvians,  the  year  began  at  the 
winter  solstice,  and  among  the  Mexicans  at  the  vernal  equinox. 
The  year  of  the  former  was  lunar,  and  was  divided  into  four 
equal  parts,  bearing  the  names  of  their  four  principal  festivals, 
instituted  in  honor  of  their  four  divinities  allegorical  of  the  sea 
sons.  The  Mexicans  had  a  year  of  360  days  and  5  supple- 
mentary days.  They  divided  it  into  18  months  of  20  days  and 
had  a  leap  year. 

Lunar  Fear.— Though  the  return  of  the  seasons  obviously 
depends  on  the  motion  of  the  sun,  or  rather  of  the  earth  in  its 
orbit,  some  nations  have  chosen  to  regulate  their  civil  year  by 
the  motions  of  the  moon;  and  many  others  have  formed  luni- 
solar  years,  by  combining  periods  determined  by  the  revolu- 
tion of  both  bodies.  The  proper  lunar  year  consists  of  twelve 
lunar  months,  or  lunations,  and  consequently  contains  only  354 
days;  its  commencement,  therefore,  anticipates  that  of  the  solar 
year  by  upwards  of  eleven  days,  and  passes  through  the  whole 
circle  of  the  seasons  in  about  34  lunar  years.  The  inconve- 
nience attending  this  circumstance  has  been  so  universally  per- 
ceived, that,  excepting  the  modern  Jews  and  Mohammedans, 
almost  all  nations  which  have  regulated  their  months  by  the 
moon,  have  employed  some  method  of  intercalation  for  the  pur- 
pose of  retaining  the  beginning  of  the  year  at  nearly  the  same 
place  in  the  seasons.  These  methods  are  founded  on  certain  luni- 
solar  periods  or  cycles,  which  were  established  in  the  most 


548  THE    PHILOSOPHY    OF    ARITHMETIC. 

ancient  times,  and  which,  with  other  relics  of  a  barbarous  age, 
are  still  preserved  in  our  ecclesiastical  calendars, 

Ecclesiastical  Calendars. — The  adaptation  of  the  civil  to  the 
solar  year  is  attended  with  no  difficulty;  but  the  church  calen- 
dar for  regulating  the  movable  feasts  imposes  conditions  less 
easily  satisfied.  The  early  Christians  borrowed  a  portion  of 
their  ritual  from  the  Jews,  whose  year  was  luni-solar.  Easter,, 
the  principal  Christian  festival,  in  imitation  of  the  Jewish 
passover,  was  celebrated  about  the  time  of  the  full  moon. 
Differences  of  opinion,  and  consequently  disputations,  soon 
arose  as  to  the  proper  day  on  which  the  celebration  should  be 
held.  In  order  to  put  an  end  to  an  unseemly  contention,  the 
Council  of  Nice  laid  down  a  specific  rule  that  Easter  should 
always  be  celebrated  on  the  Sunday  which  immediately  follows 
the  full  moon  that  happens  upon,  or  next  after,  the  day  of  the 
vernal  equinox.  In  order  to  determine  Easter  according  to 
this  rule  for  any  particular  year,  it  is  necessary  to  reconcile 
three  periods;  namely,  the  week,  the  lunar  month,  and  the 
solar  year.  To  find  the  day  of  the  week  on  which  any  given* 
day  of  the  year  falls,  it  is  necessary  to  know  on  what  day  of 
the  week  the  year  began.  In  the  Julian  calendar  this  was 
easily  found  by  means  of  a  short  period  or  cycle  of  28  years, 
after  which  the  year  begins  with  the  same  day  of  the  week. 
In  the  Gregorian  calendar  this  order  is  interrupted  by  the 
omission  of  the  intercalation  in  the  last  year  of  the  century. 
But  to  render  any  calculation  unnecessary,  a  table  is  given  in 
the  prayer-books,  showing  the  correspondence  of  the  days  of 
the  year  and  the  week  for  the  current  century.  The  connection 
of  the  lunar  month  with  the  solar  year  is  an  ancient  problem, 
for  the  resolution  of  which  the  Greeks  invented  cycles  or 
periods,  which  remained  in  use  with  some  modifications  till 
the  time  of  the  Gregorian  reformation.  The  author  of  the 
Gregorian  calendar,  Luigi  Lilio  Ghiraldi,  or,  as  he  is  frequently 
called,  Aloysius  Lilius,  employed  for  the  same  purpose  a  set  of 
numbers  called  Epacts.     It  is  to  be  desired  that  this  compli- 


MEASURES   OF   TIME.  549 

eated  system  of  rules  and  tables  were  rendered  unnecessary  by 
abolishing  the  use  of  the  lunar  month,  and  causing  Easter  to 
fall  invariably  on  the  same  Sunday  of  a  calendar  month;  for 
example,  the  first  or  second  Sunday  in  April. 

The  Measures. — Measures  of  Time  are  definite  portions  of 
duration,  fixed  by  the  revolutions  of  the  earth  on  its  axis  and 
around  the  sun.  The  unit  of  time  is  the  day,  subdivided  into 
hours,  minutes,  and  seconds.  The  other  divisions  of  time, 
arising  from  the  day,  are  weeks,  months,  and  years.  The 
length  of  the  day  is  determined  by  the  revolution  of  the 
earth  on  its  axis.  The  Sidereal  Day  is  the  exact  time  of  the 
revolution  of  the  earth  on  its  axis.  The  Solar  Day  is  the  time 
of  the  apparent  revolution  of  the  sun  around  the  earth.  The 
Astronomical  Day  is  the  solar  day,  beginning  and  ending 
at  noon.  The  Civil  Day  is  the  average  length  of  all  the  solar 
days  of  the  year;  it  begins  at  12  o'clock  midnight,  and  con- 
sists of  two  periods  of  12  hours  each. 

The  second  and  minute  are  parts  of  an  hour,  corresponding 
to  the  parts  of  a  degree  in  Circular  Measure.  Hour  is  derived 
from  the  Latin  hora,  originally  a  definite  space  of  time  fixed 
by  natural  laws ;  a  day,  derived  from  the  Saxon  daeg,  is  the 
time  of  the  revolution  of  the  earth  upon  its  axis;  a  week  is  a 
period  of  uncertain  origin,  but  which  has  been  used  from  time 
immemorial  in  Eastern  countries;  a  month,  from  monadh,  is 
the  time  of  one  revolution  of  the  moon  around  the  earth;  a 
year,  from  Saxon  gear,  is  the  time  of  the  earth's  revolution 
around  the  sun;  century  comes  from  the  Latin  centuria,  a 
collection  of  a  hundred  things. 

The  week  is  supposed,  by  some  writers,  to  be  derived  from 
a,  tradition  of  the  creation ;  by  others,  to  have  been  suggested 
by  the  phases  of  the  moon ;  while  others  refer  its  origin  to  the 
seven  planets  known  in  ancient  times.  This  opinion  explains 
the  circumstance  that  the  days  of  the  week  have  been  univer- 
sally named  after  the  planets,  in  a  particular  order.  In  the 
ancient  Egyptian  astronomy,  the  order  of  the  "  planets,"  m 


550  THE    PHILOSOPHY    OF    ARITHMETIC. 

respect  of  distance  from  the  earth,  beginning  with  the  most 
remote,  was  Saturn,  Jupiter,  Mars,  the  Sun,  Venus,  Mercury, 
the  Moon.  The  day  was  divided  into  24  hours,  and  each 
successive  hour  consecrated  to  a  particular  planet  in  the  order 
just  stated ;  and  each  day  was  named  after  the  planet  to  which 
its  first  hour  was  consecrated.  Supposing  the  first  hour  con- 
secrated to  Saturn,  he  would  also  have  the  8th,  15th,  and  22d 
hours.  The  23d  then  would  fall  to  Jupiter,  the  24th  to  Mars, 
the  1st  of  the  following  day  to  the  Sun,  from  which  it  would 
take  its  name.  By  proceeding  in  the  same  manner  it  is  found 
that  the  third  day  would  fall  to  the  Moon,  the  fourth  day  to 
Mars,  the  fifth  to  Mercury,  the  sixth  to  Jupiter,  the  seventh  to 
Venus,  and,  the  cycle  being  completed,  the  eighth  to  Saturn 
again,  and  so  continue  in  the  same  order.  The  Egyptians  are 
said  to  have  begun  their  week  with  Saturday.  The  Saxons 
seem  to  have  borrowed  the  week  from  some  Eastern  nation, 
substituting  the  names  of  their  own  divinities  for  those  of 
^he  gods  of  Greece 

January  is  derived  from  Janus,  the  old  Latin  god  of  the 
sun  and  the  year,  to  whom  this  month  was  held  sacred. 
February  is  from  februa,  the  Roman  festival  of  propitiation,  cel- 
ebrated on  the  15th  of  this  month.  January  and  February  were 
added  to  the  Roman  calendar  by  Numa,  Romulus  having  pre- 
viously divided  the  year  into  10  months.  March  is  from  Mars, 
the  god  of  war  and  reputed  father  of  Romulus.  It  was  the 
first  month  of  the  Roman  calendar.  April  is  probably  from 
the  Latin  aperire,  to  open,  from  the  opening  of  the  buds,  or 
the  bosom  of  the  earth  in  producing  vegetation.  May  is  from 
31aia,  the  mother  of  Mercury,  to  whom  the  Romans  offered 
sacrifices  on  the  first  day  of  this  month.  June  is  probably 
from  Juno,  the  sister  and  wife  of  Jupiter.  July  was  named 
by  Mark  Antony  after  Julius  Caesar,  who  was  born  in  this 
month.  It  was  previously  called  Quintilis.  August  was 
named  after  Augustus  Caesar.  It  was  formerly  called  Sextilisr 
the  sixth  month.       September,  October,  November,  December ; 


MEASURES   OF   TIME.  55L 

are  respectively  named  from  the  Latin  numerals,  septem,  octo, 
novem,  decern. 

Adjustment  of  the  Calendar. — A  True  or  Solar  year  is  the 
exact  time  in  which  the  earth  revolves  around  the  sun.  It  con- 
sists of  365  days  5  hours  48  minutes  49. T  seconds.  Now,  since 
it  is  inconvenient  to  reckon  the  fractional  part  of  a  day  each 
year,  it  is  necessary  to  arrange  a  correct  calendar  in  which  each 
3Tear  may  have  a  whole  number  of  days.  This  is  done  by  caus- 
ing some  years  to  consist  of  365  days  and  others  of  366  days. 
The  former  are  called  common  years ;  the  latter,  Bissextile  or 
Leap  years. 

The  calendar  is  reckoned  according  to  the  following  rule: — 
Every  year  that  is  divisible  by  4,  except  the  centennial,  and 
every  centennial  year  divisible  by  400,  is  a  leap  year ;  all  the 
other  years  are  common  years.  The  centennial  years  are  the 
hundredth  years,  or  those  which,  when  expressed  in  figures,  end 
in  two  ciphers.  The  reason  for  this  rule  will  now  be  ex- 
plained. 

If  we  reckon  365  days  as  1  year,  the  time  lost  in  the  calen- 
dar in  one  year  is  5  hours  48  minutes  49.7  seconds,  and  in  4 
years  is  23  hours  15  minutes  18.8  seconds,  that  is,  one  day, 
lacking  only  44  minutes  41.2  seconds;  hence  the  first  error  can 
be  corrected  by  adding  one  day  every  four  years,  making  the 
year  to  consist  of  366  days. 

If  every  fourth  year  be  reckoned  as  leap  year,  since  we  add 
44  minutes  etc.  too  much,  the  time  gained  in  the  calendar  in 
4  years  is  44  minutes  41.2  seconds,  and  in  100  years  it  will  be 
18  hours  37  minutes  10  seconds,  that  is,  one  day  lacking  5 
hours  22  minutes  50  seconds;  hence  the  second  error  may  be 
corrected  by  deducting  one  day  from  each  centennial  leap  year, 
thus  calling  each  centennial  a  common  year  of  365  days. 

Again,  if  every  centennial  year  be  reckoned  as  a  common 
year,  since  we  do  not  add  enough,  the  time  lost  in  100  years 
will  be  5  hours  22  minutes  50  seconds,  and  in  400  years  it  will 
be  21  hours  31  minutes  and  20  seconds;  hence  the  time  lost  in 


552  THE    PHILOSOPHY    OF    ARITHMETIC. 

400  years  will  be  1  day  lacking  2  hours  28  minutes  40  seconds, 
and  this  error  may  be  rectified  by  making  every  fourth  cen- 
tennial year  a  leap  year.  In  the  same  way  we  may  make  the 
calendar  correct  for  any  number  of  years. 

The  Gregorian  method  of  intercalation  thus  gives  97  inter- 
calations in  400  years ;  consequently  400  years  contain  146097 
days,  and  the  mean  length  of  the  Gregorian  year  is  365  days 
5  hours  49  minutes  12  seconds;  exceeding  the  true  solar  year 
oy  22.3  seconds  ;  an  error  which  amounts  to  one  day  in  about 
3866  years. 

If  an  astronomer  were  required,  without  any  reference  to 
established  usages,  to  give  a  rule  for  intercalation  by  which  the 
civil  year,  while  it  always  coincides  with  the  commencement  of 
a  day,  should  deviate  the  least  possible  from  the  same  instant 
of  the  solar  year,  he  would  proceed  as  follows:  The  fraction 
by  which  the  solar  year  exceeds  365  days  is  .2422414,  which, 
converted  into  a  continued  fraction,  gives  the  following  series 
of  approximations,— ±,  ■&,  &.  iWr*  AV*  tWp  Of  these  the 
first  gives  an  intercalation  of  1  day  in  4  years,  which  supposes 
the  year  to  be  365£  days.  The  second  gives  7  intercalations  in 
29  years,  and  supposes  the  length  of  the  year  to  be  365  days 
5  hours  47  minutes  35  seconds,  which  is  somewhat  too  small. 
The  third  fraction,  ^,  is  remarkable  as  giving  a  year  which 
differs  in  excess  from  the  true  solar  by  15.38  seconds,  so  that 
by  intercalating  8  times  in  33  years,  or  7  times  successively 
every  fourth  year,  and  once  at  the  end  of  the  fifth  year,  the 
difference  between  the  civil  and  solar  years  would  only  accu- 
mulate to  a  day  in  about  5600  years ;  while  in  the  Gregorian 
calendar  the  error  amounts  to  a  day  in  about  3866  years.  The 
modern  Persians  are  said,  but  not  on  very  good  authority,  to 
intercalate  in  this  manner.  Nevertheless  the  Gregorian  rule 
has  the  advantage  that  leap  year  is  always  readily  distinguished. 

Day  of  the  Week. — There  is  a  simple  method  of  finding  the 
day  of  the  week  upon  which  any  year  begins,  and  also  the  day 
of  tne  week  upon  which  any  event  has  occurred,  which  we 


MEASURES   OF    TIME.  553 

will  explain.  Take  the  first  seven  letters  of  the  alphabet,  A, 
B,  C,  D,  E,  F,  G,  A  representing  the  1st  of  January,  B  the 
2d,  C  the  3d,  etc.,  A  again  the  8th,  B  the  9th,  and  so  on 
through  the  year.  Now,  one  of  these  letters  will  stand  for 
Sunday,  and  this  letter  is  called  the  Sunday  or  Dominical 
Letter.  Thus,  if  January  begins  on  Sunday,  A  will  be  the 
dominical  letter  for  that  year ;  if  January  begins  on  Monday, 
the  first  Sunday  will  be  the  7th ;  hence  G,  the  Tth  letter,  will 
be  the  dominical  letter.  In  leap  year  we  have  two  dominical 
letters,  one  for  January  and  February,  and  the  next  preceding 
for  the  remainder  of  the  year. 

We  find  the  dominical  letter  for  any  year  according  to  the 
Julian,  or  Old  Style,  by  the  following  rule:  To  the  given  year 
add  one-fourth  of  itself  plus  4,  and  divide  the  sum  by  1.  If 
there  is  no  remainder,  the  dominical  letter  is  G;  if  I  remains, 
F;  and  so  on  in  reverse  order.  In  leap  year  the  letter  thus 
found  will  be  the  dominical  letter  for  the  last  ten  months,  and 
the  next  following  letter  for  the  first  two  months.  Having 
the  dominical  letter,  we  can  easily  find  the  day  on  which  the 
year  begins. 

When  the  year  is  reckoned  according  to  the  New  Style,  we 
find  the  domiuical  letter  by  the  following  rule:  Divide  the 
number  of  the  century  by  4,  subtract  the  remainder  from  3, 
add  twice  this  remainder  to  the  odd  years  plus  {  of  the  odd 
years,  and  divide  the  sum  by  1.  If  there  is  no  remainder, 
the  dominical  letter  is  G ;  if  1  remains  it  is  F,  etc. 

Having  found  the  day  of  the  week  on  which  January  begins 
in  any  year,  we  may  find  on  what  day  each  month  commences 
by  the  following  couplet : 

At  Dover  Dwelt  George  Brown  Esquire, 
Good  Captain  French  and  David  Friar. 
The  initial  letters  of  the  several  words  indicate  the  months  in 
their  order,  the  first  word  the  first  month,  the  second  word  the 
second  month,  etc.  Now,  if  January  begins  on  Sunday,  D, 
the  letter  for  February,  being  the  fourth  of  the  series,  indicates 
24 


554  THE    PHILOSOPHY    OF    ARITHMETIC. 

that  February  begins  on  the  fourth  day  of  the  week,  or 
Wednesday,  etc.  In  leap  year,  the  months  after  February 
come  in  one  day  later. 

The  method  may  be  illustrated  by  the  following  example : 
Upon  what  day  of  the  week  did  the  battle  of  Bunker  Hill 
occur,  June  17,1775?  Dividing  17,  the  number  of  the  century, 
by  4,  we  have  a  remainder  of  1 ;  subtracting  this  from  3  and 
multiplying  by  2  we  have  4 ;  4  plus  75,  plus  £  of  75,  omitting 
the  fraction,  gives  97,  which  divided  by  7  gives  a  remainder 
of  6  ;  hence  the  dominical  letter  is  6  before  G,  or  A.  Therefore 
the  1st  day  of  January  was  Sunday.  Now,  by  the  couplet,  the 
first  day  of  June  is  E  ;  and  as  A  is  the  dominical  letter,  June 
1st  was  on  Thursday,  and  the  17th  was  on  Saturday. 

In  closing  this  chapter,  we  call  attention  to  a  simple  point  upon 
which  there  has  been  some  popular  misapprehension — that  is, 
the  manner  of  reckoning  the  centuries.  The  centuries  are  num- 
bered from  the  beginning  of  the  Christian  era.  Thus,  all  events 
transpiring  from  the  beginning  of  this  era  until  the  end  of  the 
first  hundred  years  are  reckoned  as  belonging  to  the  first  cen- 
tury ;  all  events  between  the  end  of  the  100th  year  and  the  end  of 
the  200th  year,  are  reckoned  in  the  second  century,  etc.  It  is 
thus  seen  that  the  18th  century  closed  with  the  end  of  the  year 

1800,  and  the  19th  century  began  at  the  beginning  of  the  year 

1801.  Also  the  19th  century  ended  with  the  year  1900,  and  the 
20th  century  began  with  the  beginning  of  the  year  1901. 


CHAPTER  VI. 

THE   METRIC   SYSTEM. 

THE  Old  System  of  weights  and  measures  was  born  of  ne- 
cessity and  developed  by  chance.  Some  method  of  meas- 
uring quantity  must  have  been  coeval  with  the  race,  being  a 
necessity  of  man  as  a  social  being;  and  the  earliest  systems 
were  gradually  enlarged  and  improved  as  man  advanced  in  civ- 
ilization. Originating  by  chance  or  circumstances,  however, 
rather  than  by  science,  they  lacked  symmetry  and  precision, 
were  difficult  to  learn  and  inconvenient  to  apply. 

The  two  principal  characteristics  of  a  system  of  weights 
and  measures  are  the  units  and  their  scale  of  relations.  The 
essentials  are  that  the  units  should  be  precise  and  the  scale 
regular.  Neither  of  these  conditions  exists  in  the  old  system. 
The  earliest  of  measures  were  derived  from  variable  objects  in 
nature,  and  the  scales  of  relations  are  so  capricious  as  to  defy 
an  attempt  to  account  for  their  origin.  No  two  tables  have  the 
same  scale,  and  seldom  have  more  than  two  units  of  a  scale  the 
same  relation. 

The  units  of  the  old  system  derived  from  natural  objects,  and 
without  any  scientific  considerations,  were  necessarily  uncer- 
tain. The  foot,  among  different  nations,  would  vary  even  more 
than  the  size  of  their  real  feet,  and  the  same  lack  of  precision 
would  belong  to  all  other  measures.  But  the  scale  of  relations, 
arising  from  accidental  circumstances,  and  being  controlled  by 
no  scientific  principle,  was,  if  possible,  even  more  irregular. 
About  thirty  different  numbers  are  used  in  the  ordinary  tables 
of  the  United  States  and  England ;    and   these  are   arranged 

(555) 


556  THE    PHILOSOPHY    OF    ARITHMETIC. 

without  any  regard  to  convenience  or  law.  Such  is  the  system 
of  weights  and  measures  that  we  have  been  required  to  learn 
and  apply  to  the  business  transactions  of  life. 

Science  found  the  old  system  in  the  condition  described,  and 
endeavored  to  reform  it.  It  established  the  units  upon  scien- 
tific principles,  and  thus  gave  it  precision.  The  yard,  which 
at  one  time  was  the  length  of  the  king's  arm,  was  fixed  by  the 
vibrations  of  the  pendulum",  and  from  it  all  other  units  of  meas- 
ure and  weight  were  derived.  But  though  science  could  give 
exactness  to  the  old  system,  it  could  not  impart  to  it  simplicity 
and  regularity.  The  confusion  was  too  great  even  for  the 
touch  of  science  to  reduce  to  order.  There  was  only  one  way 
in  which  such  a  reform  could  be  accomplished,  and  that  was  to 
throw  away  the  old  system  and  construct  a  new  one.  This  was 
done,  and  the  result  is  the  Metric  System. 

History. — This  system  was  suggested  as  long  ago  as  1528, 
by  Jean  Fernal,  a  physician  of  Henry  II.,  of  France ;  but  the 
suggestion  did  not  take  a  practical  turn  until  1790,  when  Prince 
Talleyrand  distributed  among  the  members  of  the  Constituent 
Assembly  of  France  a  proposal,  founded  upon  the  excessive 
diversity  and  confusion  of  the  weights  and  measures  then  pre- 
vailing all  over  that  country,  for  the  formation  of  a  new  sys- 
tem upon  the  principle  of  a  single  and  universal  standard. 

A  committee  of  the  Academy  of  Sciences,  consisting  of  five 
of  the  most  eminent  mathematicians  of  Europe, — Borda,  La- 
grange, Laplace,  Monge,  and  Condorcet, — was  subsequently 
appointed,  under  a  decree  of  the  Constituent  Assembly,  to 
report  upon  the  selection  of  a  natural  standard ;  and  the  com- 
mittee proposed  in  its  report  that  the  ten-millionth  part  of 
the  quarter  of  the  meridian  of  Paris  should  be  taken  as  the 
standard  unit  of  lineal  measure. 

Delambre  and  Mechain  were  appointed  to  measure  an  arc 
of  the  meridian  between  Dunkirk  and  Barcelona,  as  Cassini 
had  been  appointed  in  1669.  They  commenced  their  labors  at 
the  most  agitated  period  of  the  French  Revolution.     At  every 


THE   METRIC   SYSTEM.  557 

station  of  their  progress  in  the  field-survey,  they  were  arrested 
by  the  suspicions  and  alarms  of  the  people,  who  took  them  for 
spies  or  engineers  of  the  invading  enemies  of  France.  The 
result  was  a  wonderful  approximation  to  the  true  length,  and 
one  in  the  highest  degree  "  creditable  to  the  French  astrono- 
mers and  geometricians,  who  carried  on  their  operations  under 
every  difficulty  and  at  the  hazard  of  their  lives,  in  the  midst 
of  the  greatest  political  convulsion  of  modern  times." 

The  arc  of  the  meridian  extending  from  Dunkirk  to  Barcelona, 
comprising  about  10°  of  latitude,  was  measured  trigononietri- 
cally  and  compared  with  the  arc  measured  by  Bouguer  and 
La  Condamine  in  Peru  in  1736;  and  the  length  of  the  quarter 
of  the  meridian,  or  the  distance  from  the  Equator  to  the  Pole, 
was  calculated.  This  length  was  divided  into  ten  million 
equal  parts,  and  one  of  these  parts  was  taken  for  the  unit 
of  length  and  called  a  metre,  from  the  Greek  word  ^erpov,  a 
measure.  The  distance  was  measured  in  terms  of  the  toise,  or 
old  fathom  (six-foot)  measure  of  France,  which  was  used  as 
the  measurement  of  the  base  lines;  and  its  ten-millionth 
part,  or  the  Metre,  was  determined  to  be  443.296  lines,  the 
line  being  j^  of  a  foot.  It  appears  thus  that  four  metres 
would  exceed  two  toises  by  the  19th  part  of  a  toise,  very 
nearly;  and  the  following  process  of  constructing  the  metre 
was  adopted.  Nineteen  pieces  were  made  as  nearly  as  possi- 
ble equal  to  each  other,  so  that  their  aggregate  would  be  a 
toise ;  upon  examination  it  was  found  that  one  had  almost 
exactly  the  required  length.  This  piece,  together  with  the  two 
toises  that  had  served  for  the  base  measurements,  was  placed 
in  the  comparator  and  compared  with  four  single  metre  bars 
abutted  together,  which  were  similarly  compared  with  each 
other,  and  adjusted  by  grinding  and  polishing  their  ends  until 
they  had  the  desired  length.  These  bars  were,  like  the  toises, 
of  iron;  one  of  them  was  chosen  for  the  French  standard,  from 
which  the  platinum  metre  of  the  archives,  which  is  the  legal 
standard  of  France,  was  copied.       Another  of  these  original 


558  THE    PHILOSOPHY    OF    ARITHMETIC. 

metres  was  brought  to  the  United  States  and  has  served  aa 
the  standard  for  the  geodesy  of  the  coast  survey,  and  for  the 
construction  of  a  metric  standard  for  this  country. 

If  the  arc  of  the  meridian  is  calculated  from  the  result  of 
French  researches,  the  metre  itself  is  equal,  in  English  meas- 
urement, to  39.37079  inches;  and  multiplying  this  length  by 
ten  millions,  the  length  of  the  quadrant  of  the  meridian  when 
converted  into  feet,  will  be  32,808,992  feet.  Sir  John  Herschel 
estimates  the  length  of  the  quadrant  of  the  meridian  at 
32,813,000  feet;  so  that,  according  to  his  calculation,  there  is 
a  difference  between  the  French  and  the  new  estimate  of  the 
quadrant  of  4008  feet,  and  therefore  the  French  length  of  the 
quadrant  is  81*94  too  short,  and  the  metre  is  ^g-  of  an  inch 
less  than  the  length  of  the  ten-millionth  part  of  the  quadrant. 

This  error  of  ^  °f  an  incn  in  tne  determination  of  the 
metre,  however,  supposing  it  possible  to  establish  it  absolutely, 
does  not  make  the  metric  system  less  complete  or  useful;  but 
is  more  than  counterbalanced  by  the  extreme  simplicity,  sym- 
metry, and  convenience  of  the  system.  Professor  Bessel  ob- 
served with  respect  to  the  metre,  that  in  the  measurement  of  a 
length  between  two  points  on  the  surface  of  the  earth,  there  is 
no  advantage  at  all  in  proving  the  relation  of  the  measured  dis- 
tance to  a  quadrant  of  the  meridian.  Professor  Miller,  of 
Cambridge,  also  deems  the  error  in  the  relation  of  the  metre 
to  the  quadrant  of  the  meridian  to  be  of  no  consequence.  An 
error  which  has  also  been  discovered  in  the  kilogramme  is  pro- 
nounced by  the  same  authorities  to  be  of  little  importance  in 
a  practical  point  of  view. 

It  is  difficult  to  make  an  exact  comparison  between  the  metre 
and  the  inch  or  yard,  arising  from  the  fact  that  the  metre  is  an 
end  measure  of  platinum,  having  its  standard  length  at  32°  F  , 
while  the  yard  is  a  line  measure  of  bronze,  standard  at  62°  F. 
They  cannot,  therefore,  be  directly  compared,  and  the  dilatation 
by  temperature  comes  into  eifect,  and  requires  to  be  ascertained 
with   the  utmost   accuracy.      The   means  of  comparison    for 


THE   METRIC    SYSTEM.  559 

standards  of  length  are  different  according  to  their  being 
line  or  end  measures.  In  the  former  case,  when,  as  in  the 
British  yard,  the  standard  length  is  contained  between  lines 
drawn  upon  the  bar,  the  comparator  is  necessarily  optical, 
which  enables  us  to  measure  by  means  of  micrometer  micro- 
scopes the  minute  differences  between  different  measures 
traced  from  the  same  standard  by  mechanical  means.  But  when 
the  standards  are  end  measures,  or  contained  between  the  ter- 
minal planes  of  the  bar,  the  comparison  is  necessarily  made  by 
actual  contact,  the  rotation  of  a  mirror,  or  tilting  of  a  delicate 
level,  being  used  as  the  means  of  indicating  the  minute  differ- 
ences. In  standard  measures  of  the  latter  kind,  it  is  now 
usual  to  make  the  terminal  surfaces  very  small,  and  ground  off 
parallel  to  each  other  by  means  of  cylindrical  bearings  near 
each  end  of  the  bar.  It  is  only  by  such  means  that  parallelism 
approaching  to  geometrical  accuracy  can  be  obtained.  In  both 
kinds  of  comparison,  a  precision  of  the  x^nnnnr  Part  of  an  mcn 
may  be  reached.  The  greatest  difficulty  in  obtaining  extreme 
precision  arises  from  the  variability  of  temperature ;  and  this 
is  greatly  enhanced  when  the  measures  compared  are  of  differ- 
ent volume,  and  still  more  when  of  different  metals.  In  com- 
parisons of  precision,  it  is  therefore  necessary,  to  insure  a  great 
uniformity  of  temperature,  to  prevent  as  much  as  possible  the 
influence  of  the  bodily  heat  of  the  observer  upon  the  appar- 
atus. 

The  System. — The  Metric  System  is  so  called  because  the 
base,  or  fundamental  unit  from  which  all  the  other  units  are 
derived,  is  the  Meter.  The  units  of  the  different  measures  have 
simple  and  definite  relations,  and  the  whole  system  is  founded 
upon  the  decimal  scale.  A  unit  of  any  measure  being  estab- 
lished, the  other  denominations  are  derived  by  taking  decimal 
multiples  and  divisions  of  the  unit.  The  multiples  are  named 
by  prefixes  derived  from  the  Greek, — deka,  ten  ;  hecto,  hundred,, 
etc.;  those  denoting  divisions,  from  the  Latin, — deci,  tenth,  etc. 
The  scale  ascends  and  descends  by  tens,  the  same  as  our  or 


660  THE    PHILOSOPHY    OF    ARITHMETIC. 

dinary  scale  of  numeration  and  notation.  Any  quantity  con- 
sisting of  several  denominations  is  thus  written  and  treated 
like  an  integer  and  decimal,  the  decimal  point  separating  the 
unit  and  its  divisions.  Another  merit  of  the  system  is  that 
the  units  are  all  correlated,  being  derived  from  the  standard 
unit  or  Meter. 

The  Unit  of  Length  is  the  Meter.  It  equals  39.37  inches, 
which  is  in  theory  the  ten-millionth  part  of  a  degree  of  latitude, 
being  a  little  longer  than  the  yard,  the  present  unit  of  length. 
Another  expression  for  its  length  which  may  be  easily  remem- 
bered, is  3  feet,  3  inches,  and  3  eighths  of  an  inch.  The  Meter 
is  the  unit  from  which  all  the  other  units  are  derived.  It  is 
the  basis,  the  fundamental  unit,  of  the  whole  system.  Ordi- 
nary lengths  are  measured  by  the  meter ;  very  small  distances 
by  the  millimeter  (y^Vff  of  a  meter),  and  long  distances  by  the 
kilometer  (1000  meters).  The  five-cent  piece,  adopted  in  1866, 
is  -^  of  a  meter  or  2  centimeters  in  diameter.  A  decimeter  is 
about  4  inches  ;  a  millimeter,  about  -fa  of  an  inch  ;  a  kilometer, 
about  200  rods,  or  f  of  a  mile. 

The  Unit  of  Surface,  used  in  measuring  land,  is  the  Are, 
which  is  a  square  decameter.  It  equals  3.9574  perches,  or 
0.0247  acre.  The  are,  centiare,  and  hectare  are  the  denomina- 
tions principally  used,  as  they  are  exact  squares ;  the  deciare 
is  not  a  square, but  merely  the  tenth  of  an  are,  and  the  decare 
is  merely  ten  ares.  The  centiare  is  a  square  meter.  A  hect- 
are is  nearly  2J  acres ;  an  acre  is  nearly  40  ares.  Other  sur- 
faces, as  cloth,  lumber,  paper,  etc.,  are  measured  by  the  square 
meter. 

The  Unit  of  Volume  used  in  measuring  wood,  is  the  Stere, 
which  is  a  cubic  meter.  The  stere,  decistere,  and  decastere  are 
principally  used.  3.6  steres  very  nearly  equal  a  cord.  Other 
solid  bodies,  like  stone,  sand,  gravel,  etc.,  are  measured  by  the 
cubic  meter  and  its  divisions. 

The  Unit  of  Capacity  is  the  Liter,  which  equals  a  cubic 
decimeter,  and  contains  61.027  cubic  inches,  or   2.1135   pints 


THE    METRIC    SYSTEM.  561 

wine  measure,  or  1.816  pints  dry  measure.  This  measure  is 
used  both  for  measuring  liquids  and  dry  subtances.  Liquids 
are  usually  measured  by  the  liter,  and  grain  by  the  hectoliter, 
which  equals  nearly  2  J  bushels,  or  £  of  a  barrel.  4  liters  are 
a  little  more  than  a  gallon,  and  35  liters  nearly  a  bushel. 

The  Unit  of  Weight  is  the  Gram.  It  is  the  weight  of  a 
cubic  centimeter  of  distilled  water  at  the  temperature  of  melt- 
ing ice,  and  equals  15.432  Troy  grains.  It  is  used  in  weighing 
letters,  in  mixing  and  compounding  medicines,  and  in  weigh- 
ing all  very  light  articles.  The  five-cent  coin  adopted  in  1866, 
weighs  5  grams.  The  kilogram,  usually  abbreviated  into  kilo, 
is  the  ordinary  unit  of  weight.  It  equals  about  2-J-  pounds 
Avoirdupois.  Meat,  sugar,  etc.,  are  bought  and  sold  by  the 
kilogram.  In  weighing  heavy  articles,  two  other  weights,  the 
quintal  (100  kilograms)  and  the  tonneau  (1000  kilograms)  are 
used.  The  U.  S.  Post-office  receives  15  grams,  though  a  little 
over  weight,  as  equivalent  to  an  ounce  Avoirdupois. 

Its  Adoption. — The  Metric  System  was  adopted  in  France 
in  U95,  but  was  tardily  accepted  by  the  people.  Though  it 
was  early  made  compulsory,  it  became  necessary  to  relax  the 
law  so  as  to  permit  the  use  of  halves  and  quarters  of  the  sev- 
eral units..  Since  1840,  however,  the  metric  measures  have 
been  the  only  ones  in  common  use  in  France  ;  and  the  system 
has  found  a  very  large  acceptance  elsewhere.  It  has  been 
adopted  by  Italy,  Spain,  Portugal,  Greece,  Belgium,  Holland, 
and  partially  by  Denmark  and  Switzerland.  Many  of  the 
German  states  have  also  expressed  their  approval  of  the  system, 
and  the  half  kilogram  has  been  introduced  into  all  the  great 
mercantile  operations  in  Austria.  In  1863,  at  the  International 
Statistical  Congress  held  at  Berlin,  a  resolution  was  passed 
recommending  the  metric  system  as  the  most  convenient  for 
international  measures.  A  Commission  of  the  Imperial  Acad- 
emy of  Sciences  in  St.  Petersburg  has  recommended  that  such 
alterations  be  made  in  Russian  weights  and  measures  as  would 
put  them  in  conformitv  with  the  French  system. 
36 


562  THE    PHILOSOPHY    OF    ARITHMETIC. 

In  England,  the  metric  system  has  been  extensively  used 
among  scientific  men  for  many  years ;  and  in  1864  the  Parlia- 
ment passed  an  act  legalizing  its  use  throughout  the  British 
Empire.  In  18GG,  the  Congress  of  the  United  States  author- 
ized its  use  in  this  country  ;  and  to  facilitate  its  introduction 
directed  that  the  new  five-cent  piece  should  weigh  five  grams 
and  be  one-fiftieth  of  a  meter  in  diameter.  Owing  to  the  com- 
position of  the  alloy,  it  was  found  necessary  to  make  its  diam- 
eter a  little  greater  than  one-fiftieth  of  a  meter;  48.6  nickel 
five-ceut  pieces,  laid  side  by  side,  measure  one  meter.  It  was 
also  ordered  that  letters  in  post-offices  should  be  weighed  by 
the  gram,  but  this  latter  provision  has  not  been  carried  out. 
There  is,  however,  a  strong  feeling  among  scientific  men  in 
favor  of  the  system;  and  the  time  is  not  very  far  distant  when 
it  will  be  universally  used  in  this  country.  And  more  than 
this, — to  the  honor  of  France  and  the  scientific  men  who  sug- 
gested and  perfected  it, — the  metric  system  is  destined  to  be- 
come the  system  of  the  civilized  world. 


INDEX 


Abacus,  29,  145,  159  ;  description  and 
use  of,  148,  152. 

Abstract  numbers,  18,  79,  80. 

Account,  auditor's,  158  ;  merchant's, 
15S. 

Adams,  Mr.,  36S. 

11       John  Quincy,  504. 

Addition,  rule  in  Lilawati,  44 ;  Greek, 
13S ;  palpable,  156 ;  reasoning  in, 
177 ;  discussion  of,  207  ;  principles, 
207;  cases,  208;  rule,  211. 

Adjustment  of  calendar,  551. 

Airy,  503. 

Alligation,  in  the  Lilawati,  332 ;  by  the 
Arabians,  333  ;  by  the  Italians,  333. 

Almanac,  supposed  to  be  by  Roger 
Bacon,  26 ;  in  Bennet  College,  27. 

Alphonso,  king  of  Castile,  27. 

Alsted,  449. 

American  measures,  504;  weights, 
515. 

Amicable  numbers,  375,  3S3,  3S4. 

Analysis,  15,  17;  arithmetical,  1S5 ; 
syllogistic,  191. 

Apollonius,  136,  137,  139,  140. 

Apothecaries'  weight,  519. 

Approximate  roots,  282. 

Arabian  methods  of  multiplication, 
48  ;  extraction  of  square  root,  58  ; 
cube  root,  60. 

Arabic  characters  not  invented  by 
Arabs, 23  ;  introduced  into  Europe, 
24  ;  two  sets  of,  29  ;  method  of  no- 
tation, 101  ;  origin  of,  104. 

Arago,  297,  300. 


Archimedes,  33,  136,  137,  139,  140, 
297,  371,  439. 

Aristotle,  13,  SI,  122,  371. 

Arithmetic,  logical  outline  of,  13, 19 ; 
philosophy  of,  14 ;  science  of,  13 ; 
origin  and  development  of,  21 ;  in- 
troduction into  Europe,  27;  early 
writers  on,  32 ;  subject  matter  of, 
67. 

Arithmetical,  origin  of  processes,  41 ; 
language,  41 ;  symbols,  origin  of, 
108 ;  reasoning,  165,  173 ;  ideas, 
173 ;    axioms,   173 ;   analysis,   185. 

Arithmetical  progression,  342  ;  nota- 
tion, 342 ;  cases,  342 ;  method  of 
treatment,  343  ;  history,  343. 

Arya-bhatta,  44. 

Aviccnna,  34. 

Avoirdupois  weight,  519. 

Awgrym  stones,  146. 

Axioms,  arithmetical,  173  ;  In  reason- 
ing, 174. 

Babbage,  calculating  machine,  161. 

Bachct,  372. 

Bacon,  Roger,  almanac  ascribed  to 

him,  26. 
Baily,  503. 
Baker,  II.,  37,  330. 
Barlow,  Theory  of  Xumbers,  138, 140, 

373,  3S2,  3S4,  387. 
Barrow,  Dr.,  edition  of  Euclid,  33; 

S2,  297. 
Basis  of  the  scale  of  numeration,  113. 
Bede,  Venerable,  153,  344. 


(5633 


564 


INDEX. 


Beginning  of  year,  546. 

Bernoulli,  John,  435. 

Bernoullis,  the,  297. 

Bessel,  558. 

Beyern,  Johann  Hartman,  447. 

Bhascara,  43. 

Binary  scale,  116, 120. 

Biot,  300,  303. 

Bird,  Mr.,  502. 

Boethius,  24,  29,  31,  34,  53. 

Bonacci,  Leonardo,  or  Leonard  of 
Pisa,  27,  36,  52,  2S2,  284. 

Book-keeping  by  double  entry,  earli- 
est work  on,  38. 

Borda,  556. 

Borgo,  Peter,  35. 

Bouguer,  557. 

Bourdon,  297,  300,  303,  346. 

Bouvet,  121. 

Bowditch,  297. 

Brahmegupta,  44. 

Bridges,  Henry,  57 ;  Noah,  449. 

Briggs,  449,  450,  452,  453. 

Brouncker,  Lord,  435. 

Bungus,  Petrus,  87. 

Calandri,  Philip,  24,  35. 

Calendar,  542 ;  Julian,  544 ;  Gregor- 
ian. 545;  French  revolutionary,  545 ; 
ecclesiastical,  548;  adjustment  of, 
551. 

Cardan,  Jerome,  35,  284. 

Carnot,  297,  300. 

Casting  out  9's,  48,  392. 

Cataldi,  38,  434. 

Cataneo  Sienese,  Pietro,  52. 

Cauchy,  373. 

Caxton,  Wm.,  28. 

Change  of  scale,  118. 

Charles  XII.  of  Sweden,  119. 

Chinese  ideas  of  numbers,  S6  ;  ancient 
binary  system,  120;  digital  arith- 
metic, 153  ;  swan-pan,  158. 


Cipher,  22 ;  origin,  23,  107 ;  mystery 
attached  to,  107. 

Circulates,  nature  of,  460 ;  origin, 
460  ;  notation,  461 ;  definition,  462; 
cases,  463 ;  method  of  treatment, 
463  ;  treatment  of,  464 ;  reduction, 
465 ;  divisor  and  multiple,  468 ; 
principles,  470 ;  new  form,  481 ; 
complementary  repetends,  476. 

Clavis  Mathematics,  110,  448,  452. 

Clavius,  42,  43. 

Cocker's  arithmetic,  337. 

Colburn,  Warren,  166. 

Comparison,  15, 17 ;  of  integers,  186 ; 
of  fractions,  187;  direct,  192;  in- 
troduction to,  291. 

Complementary  repetends,  476. 

Composition,  240 ;  method  of  treat- 
ment, 241 ;  cases,  240. 

Compound  proportion,  318. 

Comte,  300. 

Concrete  numbers,  78,  79,  80. 

Condorcet,  556. 

Conjoined  proportion,  322. 

Continued  fractions,  434 ;  origin,  434 , 
treatment,  435. 

Counters,  146,  155. 

Court  of  Exchequer,  ancient  mode  of 
reckoning,  151. 

Cova,  120. 

Cube  root,  analytic  method,  274; 
synthetic  method,  275  ;  new  meth- 
od, 278. 

Cubic  measure,  507. 

Cubing  numbers,  analytic  method, 
265  ;  synthetic  method,  265. 

Da  Vinci,  Leonardo,  109. 
Davis,  Dr.,  109. 
Days  of  the  week,  552. 
De  Algorismo,  tract  containing  expla- 
nations of  numerals,  28. 
Decimal  point,  origin  of,  450. 


INDEX. 


565 


Decimals,  origin  of,  443  ;  history,  444 ; 
treatment  of,  455  ;  notation,  455  ; 
cases,  456;  method  of  treatment, 
456  ;  numeration,  457  ;  fundamen- 
tal operations,  459. 

Deductive  method  of  treatment  of 
fractions,  429. 

Dee,  John,  36. 

Delambre,  139,  495,  556. 

De  Morgan,  Arithmetical  Books,  40 ; 
34,  37,  39,  109,  281,  444,  446,  451, 
452,  453,  454, 

Denary  scale,  123. 

Denominate  numbers,  77 ;  nature  of, 
489 ;  definition,  490 ;  unit  of  meas- 
ure, 491 ;  quantities,  491 ;  standard 
units,  493  ;  scales,  495. 

Derivative  operations,  introduction  to, 
237. 

Descartes,  110,  262,  385. 

Devanagari  character,  22. 

Diamond  weight,  520. 

Di  Borgo,  Lucas,  or  Lucas  Pacioll,  27, 
34,  42,  50,  51,  52,  54,  56,  59,  88,  89, 
282,  284,  328,  329,  336,  337,  344, 
367,  418,  419. 

Digital  arithmetic,  153. 

Diophantus,  34, 136,  200,  335,  372. 

Dirichlet,  373. 

Disme,  La,  37,  445,  446. 

Divisibility  of  numbers,  389 ;  by  nine, 
392 ;  by  eleven,  393 ;  by  seven,  397 ; 
by  other  numbers,  401. 

Division,  rule  in  Lilawati,  49 ;  Italian 
methods,  54 ;  Greek,  139 ;  palpable, 
157 ;  reasoning  in,  181 ;  discussion  i 
of,  227 ;  priueiples,  228  ;  cases,  230 ; 
methods,  231 ;  rule,  233. 

Dodd,  Prof.,  201. 

Dominical  letter,  553. 

Dry  measure,  510. 

Du  Cange,  27. 

Duodecimal  scale,  116,  124,  126 ;  nu 
meration,  126 ;  notation,  128  ;  fun 
damental  operations,  130. 


Early  writers  on  arithmetic,  32. 

Eon  Younis,  42. 

Eisensteiu,  373. 

English    standard    measures,    500; 

weights,  514. 
English  money,  538. 
Equation  in  arithmetic,  193. 
Eratosthenes,  sieve,  33,  378. 
Essentials  of  a  base,  115. 
Even  and  odd  uumbers,  375. 
Euclid,  33,  297,  371 ;  definition  of 

number,  73. 
Euler,  201 ,  372,  373, 380, 381, 382, 384, 

385. 
Eutocius,  136,  138,  139. 
Evans,  Alexander,  284. 
Evolution,  267;  symbol,  267;  cases, 

26S  ;  extraction  of  square  root,  269 ; 

extraction  of  cube  root,  272. 

Factoring,  244;  cases,  245;  method 
of  treatment,  245 

Fermat,  200,  201,  372,  388 ;  theorem, 
380,  474. 

Fernal,  Jean,  556. 

Figurate  numbers,  385. 

Fludd,  Robert,  39. 

Fractions,  15, 76 ;  comparison  of,  187 ; 
nature  of,  413 ;  definition  of,  414 ; 
notation,  415;  history,  417;  iu  Lila- 
wati, 417 ;  classes  of  common,  420  ; 
improper,  421 ;  complex,  423  ;  treat- 
ment, 426;  cases,  426;  methods, 
428 ;  fundamental  principles,  431 ; 
continued,  434. 

French  money,  540. 

Fundamental  operations,  origin  of, 
43  ;  by  the  duodecimal  scale,  130 ; 
discussion  of,  207 ;  of  fractious,  459. 

Galley  or  scratch  method  of  division, 

54,  55,  56. 
Ganesa,    Commentary  of,  modes  of 

multiplication,  48. 
Garth.  45. 


566 


INDEX. 


Gatterer,  21. 

Gauss,  200,  373,  381,  382. 

Gellibrand,  450. 

Gemma  Frisius,  419. 

Geometrical  progression,  345;  nota- 
tion, 345 ;  rate,  345 ;  cases,  346 ; 
treatment,  347. 

Gerbert,  6aid  to  have  introduced 
Arabic  characters,  26,  29. 

German  money,  540. 

Girard,  Albert,  37,  39,  43,  111,  448, 
452. 

Gobar  figures,  30. 

Goldstucker,  99. 

Graham,  George,  501,  502. 

Greatest  common  divisor,  249 ;  cases, 
250  ;  explanation,  252 ;  abbreviated 
method,  255. 

Greek  arithmetic,  135. 

Gregorian  calendar,  545. 

Gunter,  Edmund,  453,  506. 

Hamilton,  Sir  William  Rowan,  Calcu- 
lus of  Quaternions,  70. 

Harriot,  111. 

Hartwell,  36. 

Hassler,  Prof.,  504. 

Hatton,  Edward,  36. 

Henkle,  Prof.,  method  of  naming 
higher  groups,  100 ;  views  on  ratio, 
303. 

Herigone,  Peter,  448. 

Herschel,  297,  503,  558. 

Hill,  Dr.,  334. 

Hindoo  arithmetic,  22. 

History  of  money,  529  ;  Lydian,  530  ; 
Greek,  580;  Roman,  530;  British 
and  Saxon  money,  531 ;  early 
English  money,  532 ;  of  measures 
of  time,  542 ;  of  metric  S3'stem,  556. 

Horner's  method,  281. 

Humboldt,  98. 

Hunt,  Nicholas,  39. 


Huswirt,  John,  35. 
Huygens,  435,  500. 

Induction  in  arithmetic,  197. 
Inductive  method  of  treatment  of 

fractions,  428. 
Infinite  series,  347. 
Ingpen,  Wm.,  89. 

Integers,  15,  76 ;  classes  of,  78 ;  com- 
parison of,  1S6. 
Interest,  nature  of,  361 ;  6imple,  361 ; 

rates,  363 ;  history  of,  364  ;  origin 

of  methods,  367. 
Involution,  261 ;  symbol,  261  ;  cases, 

262  ;  squaring  numbers,  263  ;  cubing 

numbers,  264. 
Italian  methods  of  multiplication,  50, 

51,  52  ;  of  division,  54. 

Jacobi,  373. 

Japanese  calculation,  159. 
Jarvis,  James,  535. 
Jefferson,  Thomas,  534. 
John  of  Halifax,  26. 
Johnson,  John,  448,  449,  453. 
Julian  calendar,  544. 

Kaestner,  43. 
Kater,  Capt.,  502,  503. 
Kavauagh,  453. 
Kelly,  Dr.,  516. 
Kepler,  450. 
Kircher,  Father,  27. 
Kobel,  Jacob,  35. 

La  Condamine,  557. 

Lacroix,  299,  300. 

Lagrange,  200,  201, 297, 298, 300, 372, 

373,  556. 
Lamy,  220. 

Laplace,  297,  298,  300,  556. 
Lauren  berg,  53. 
Lax,  Caspar,  35. 
Lazesio,  Francisco  Feliciano  da,  32S. 


INDEX. 


567 


Least  common  multiple,  257;  cases, 
258. 

Legendre,  200,  297,  298, 300, 372, 373, 
381. 

Leibnitz,  121, 161,  297. 

Leslie,  Philosophy  of  Arithmetic,  22, 
23,  26,  61,  108,  141,  147,  159. 

Libri,  110. 

Lilawati,  22  ;  story  of,  43 ;  rule  for 
addition,  44  ;  subtraction,  45 ;  mul- 
tiplication, 46  ;  division,  49  ;  powers 
and  roots,  57,  58 ;  proportion,  326 ; 
alligation,  332 ;  position,  334 ;  frac- 
tions, 417. 

Liquid  measures,  508. 

Lo-chou  and  Ho-tou,  86. 

Logical  outline  of  arithmetic,  13,  19. 

Long  measure,  505. 

Lubbock,  503. 

Lunar  year,  547. 

Lyte,  Henry,  447. 

Mabillon,  Father,  27. 

Manor an jana,  335. 

Mansel,  167,  169,  317. 

Measures  of  extension,  497;  origin 
and  history,  497 ;  English  standard 
measures,  500 ;  American  measures, 
504 ;  of  weight,  512 ;  ancient  En- 
glish measures.  517 ;  American 
system,  517 ;  of  value,  521  ;  origin, 
521 ;  of  time,  541 ;  origin,  541. 

Measurement  of  arc  of  meridian,  493, 
556. 

Mechain,  measurement  of  meridian, 
493,  556. 

Medial  proportion,  323. 

Mellis,  John,  36,  37,  337. 

Meter,  493,  559. 

Method  of  naming  numbers,  94  ;  util- 
ity of  method,  97. 

Metius,  Adrian,  439,  449. 

Metris  system,  555 ;  disadvantages  of  I 


old  system,  555  ;  history  of  metric 

system,  556 ;  units  of,  560  ;  adoption, 

561. 
Miller,  558. 

Mohammed  ben  Musa,  41. 
Money,  history    of,  529;    definition, 

533  ;  United  States  money,  533. 
Monge,  556. 

Morris,  Gouverneur,  534. 
Morris,  Robert,  534. 
Mouton,  500. 
Mungo  Park,  122. 
Muller,  29,  106,  107. 
Multiplication  is  mie  vexation,  38 ;  as 

found  in  Lilawati,  46,  47 ;  Gane6a, 

48;    Italian  methods,   50,  51,   52; 

sluggard's  rule  for,  53 ;  Greek,  138  ; 

palpable,  157;    reasoning  in,  ISO; 

discussion  of,  221 ;  principles,  222 ; 

cases,  223  ;  method,  221 ;  rule,  226. 
Multiplication  table,  ignorance  of,  53. 

Napier,  38,  447,  448,  450,  451,  452  ; 
rods,  48,  160. 

Neo-Pythagoreans,  29,  30. 

New  circulate  form,  481 ;  origin,  482 ; 
meaning,  483  ;  value,  484. 

Newton,  definition  of  number,  72; 
110 ;  binomial  theorem,  199 ;  297, 
298. 

Nicomachus,  33. 

Nonary  scale,  123. 

Norton,  Richard,  446,  447. 

Notation,  or  the  writing  of  numbers, 
101 ;  relation  to  numeration,  102 ; 
relation  to  the  base,  103;  by  the 
duodecimal  scale,  128  ;  Greek,  135. 

Number  nine,  properties  of,  404 ;  di- 
visibility by,  392. 

Number,  the  subject  matter  of  arith- 
metic, 67  ;  definition  of,  72  ;  classea 
of,  76 ;  origin  of,  67 ;  of  characterp, 
104. 


568 


INDEX. 


Numbers  expressed  by  letters  of  a 
worn,  S5. 

Numeration,  or  the  naming  of  num- 
bers, 93  ;  by  the  duodecimal  6cale, 
126;  of  decimals,  457. 

Numerical  ideas  of  the  ancients,  81 ; 
of  the  Chinese,  86. 

Objections  to  a  decimal  basis,  113. 

Octary  scale,  116,  123. 

Oldcastle,  Hugh,  38. 

Origin  of  arithmetic,  21 ;  characters, 
22,  105;  cipher,  23;  arithmetical 
characters,  41 ;  number,  67  ;  names 
of  numbers,  97 ;  Arabic  system  of 
notation,  104;  arithmetical  symbols, 
10S :  Roman  characters,  142 ;  mea- 
sures of  extension,  497 ;  measures 
of  weight,  512 ;  measures  of  value, 
521 ;  measures  of  time,  541. 

Orontius  Fineus,  46,  53, 283,  284,  444. 

Ortega,  Juan  de,  42,  283. 

Other  scales  of  numeration,  116,  120. 

Oughtred,  39, 110, 112,  331,  448,  452, 
454. 

Pacioll,  see  Di  Borgo. 

Palpable  arithmetic,  147;  notation, 
155  ;  addition,  156 ;  subtraction, 
156;  multiplication,  157;  division, 
157. 

Paper  money,  526. 

Pappus,  136. 

Parenthesis,  111. 

Partitive  proportion,  321. 

Partnership  and  barter,  337. 

Pascal,  inventor  of  arithmetical  ma- 
chine, 161. 

Peacock,  Dr.,  39,  42,  55,  61, 147,  328, 
446,  449,  450,  454. 

Pell,  Dr.  John,  110. 

Pelletier,  59,  283. 

Percentage,  nature  of,  355 ;  quanti- 
ties, 356 ;    cases,  357  ;    treatment, 


358  ;  formulas,  359 ;  applications, 
360. 

Perfect,  imperfect,  etc.,  numbers,  383. 

Perkins,  281,  478. 

Petrarch,  28. 

Philolaus,  85. 

Picard,  500. 

Pierce,  297,  298. 

Pisa,  Leonard  of,  see  Bonacci. 

Planudes,  Maximus,  45, 49,  52, 54, 61 

Plato,  13,  81,  82. 

Playfair,  502. 

Poinset,  373. 

Polyglot  numbers,  387. 

Position,  in  the  Lilawati,  334;  in 
Manoranjana,  335 ;  in  Arabian 
writers,  336;  in  Italian  writers, 
336. 

Po  were  and  roots,  rule  in  Lilawati,  57; 
Arabian  methods,  58,  60 ;  method 
of  earlier  European  mathematicians, 
59. 

Prime  and  composite  numbers,  378. 

Prinseps,  106. 

Progressions,  arithmetical,  341 ;  geo- 
metrical, 345. 

Proportion,  nature  of,  305 ;  notation 
of,  305  ;  kinds  of,  307 ;  principles, 
307  ;  demonstration,  308  ;  applica- 
tion of  simple,  310 ;  position  of  un- 
known quantity,  311 ;  symbol  for 
unknown  quantity,  312 ;  method 
of  statement,  313 ;  three  terms 
statement,  313 ;  cause  and  effect, 
314 ;  inverse,  316 ;  distinctly  arith- 
metical, 317 ;  compound,  318 ; 
principles,  319 ;  partitive,  321 ;  ori- 
gin, 321  ;  cases,  321 ;  method  of 
treatment,  322;  conjoined,  322; 
method  of  treatment,  323  ;  medial, 
323  ;  origin,  324 ;  cases,  324 ;  method 
of  treatment,  324  ;  history,  326. 

Ptolemy,  137. 

Pyramidal  numbers,  388. 


INDEX. 


569 


Pythagoras,  13,  24,  199,  297,  371; 
classes  of  numbers,  32 ;  properties 
of  numbers,  81,  82. 

Quantities,  492. 
Quantity  of  magnitude,  489. 
Quaternary  scale,  122. 
Quinary  scale,  122. 

Rabdologia,  160,  447,  451. 
Radical  sign,  111. 
Ramus,  Peter,  46,  56,  445. 
Ratio,  nature  of,  294 ;  method  of,  295; 
•«  English      method,"        "  French 
method,"    295;    reasons   for    old 
method,  295;    origin  of  symbol, 
298;     French  method  inappropri- 
ately named,  299. 
Ray,  Dr.,  299. 

Reasoning  in  arithmetic,  165 ;  nature 
of,  170 ;  in  fundamental  operations, 
177. 
Recorde,  Robert,  36,  39,  43,  53, 56, 57, 

111,  283,  330,  337,  444. 
Regiomontanus,  35,  56,  59,  444. 
Rigaud,  109. 
Risen,  Adam,  36. 
Ritchie,  109. 
Robinson,  H.  N.,  315. 
Roman  arithmetic,  141 ;    characters, 
origin  and  modification  of,  140;  com- 
putations by  pebbles,  155. 
Rudolph,  Christopher,  36, 110,  385. 
Rule  of  Three,  326. 
Sanskrit  digits,  22  ;  origin  of  numer- 
als, 99. 
Saunderson,  159. 

Scales  of  denominate  numbers,  495. 
Scheutz,    calculating  machine,  161 

162. 
Schoner,  John,  35. 
Science  of  arithmetic,  13. 
Senary  scale,  123. 


Septenary  scale,  123. 
Sheepshanks,  503. 
Shepherd,  503. 

Signs  of  addition  and  subtraction, 
109;  sign  of  multiplication,  110; 
of  division,  110 ;  of  equality,  lit ; 
of  ratio,  111,  298 ;  of  equal  ratios, 
112, 305 ;  of  inequality,  112. 
Sluggard's  rule    for   multiplication, 

53. 
Smith,  H.  J.  S.,  373. 
Square  root,  analytic  method,  270; 

synthetic  method,  271. 
Squaring  numbers,  analytic  method, 

263  ;  synthetic  method,  264. 
Stevinus,  Simon,  37,  42,  56,  283,  337, 

368,  444,  445,  446,  447,  448,  450. 
Stifel,  Michael,  36,  39,  56,  61,88, 109, 

110,  111,  418. 
Substitution,  195. 

Subtraction,  mode  of  Planudes,  45  ; 
mode  of  Ramus,  46 ;    Greek,  138 ; 
palpable,  156;  reasoning  in,  179; 
discussion    of,   213;    cases,   213; 
principles,  214 ;  methods,  215 ;  rule, 
218. 
Surface  measure,  507. 
Swan-pan,  158. 
Sylvester,  440. 

Symbols  of  number,  108 ;    of  opera- 
tion, 109 ;    of  relation,  111 ;    of  in- 
volution, 261 ;  of  evolution,  267. 
Synthesis,  15, 16. 
System  of  exponents,  110. 

Taaut  or  Thaut,  21. 

Talleyrand,  556. 

Tallies,  154. 

Tartaglia,  Nicolas,  37,  50,  52, 55,282, 
283,  284,  328,  329,  331,  333,  334, 
336,  337,  366,  367,  419,  444. 

Ternary  scale,  122. 

Thales,  33. 


570 


INDEX 


Thcon,  336. 

Theory  of  numbers,  371. 

Tucuth,  81. 

Thirtie  daies  hath  September,   etc., 

38. 
Time,  CS,  09,  70 ;    measures  of,  541, 

541). 
Tonstall,  Bishop,  42,  45,  56,  60. 
Tooke,  534. 
Transposition,  196. 
Troughton,  505. 
Troy  weight,  518. 

Unit,  15 ;  basis  of  analysis,  1S5 ;  of 
measure,  491 ;  standard  units,  493. 

United  States  money,  533  ;  history  of, 
533,  534,  537. 

Usury,  origin  and  history,  365. 

Utility  of  Arabic  method,  102. 

Van  Schooten,  E.,  3S3,  3S4. 
Vernier,  method  of  ratio,  300. 


Vicenary  scale,  125. 
Vieta,  author  of  vinculum,  111. 
Vinculum,  origin  of,  111. 
Vossius,  Gerard,  27. 

Wallis,  56,  435,  444,  449,  452. 

Webster,  William,  40,  448,  453. 

Whetstone  of  wit,  36,  ill,  444. 

Whcwell,  167,  177,  179,  201,  210. 

Wilson's  theorem,  381,  3S2. 

Wingate's  arithmetic,  453. 

Witt,  Richard,  446,  452. 

Woepcke,  modern  view  of  the  Intro- 
duction of  Arabic  characters,  29, 
31. 

Wollaston,  502. 

Wood,  Prof.,  325,  334. 

Young,  Dr.,  502. 
Young,  J.  R.,  2S1. 

Zero,  origin  of  name,  23. 


APPENDIX. 


HENKLE'S  NAMES  OF  PERIODS. 
Millions  (1),  Billions  (2),  Trillions  (3),  Quadrillions  (4), ,  Qulntilliotis  (5), 
Sext  ions  (6),  Septillions  7),  Octillions  (8),  Nonillions  (9),  DecilltonsOU), 
Undecimons  11),  Duodecillions  (12),  Tertio-deci  11  ons  (13) .<*j^<*W- 
ions  (14),  Quinto-decillions  (15),  Sexto-decillions  (16),  Octo-decill.ons  (IS), 
Nono-declUons(ll»),Vigillions  (20),  Prlmo-vigillions  (21),  Secundo-vigill. 
ions  (22),  Tertio-vitf  lions  (23),  Quarto-vigillions  (24),  Quinto-vigilhons  (25), 
Sexto-^lHons  (26),  Septo-vigillions  (27),  Octo-vigillions  (*^«J«g}- 
Millions  (29),  Trigillions  (30),  Quadragillions  (40),  Qumquagil  ons,  (50), 
K^llions;(60),t'Septua^illions  (70),  Octogillions  (80),  Nonagill.ons  90), 
Centulions  (100)  Pri.no-centillions  (101),  Deciino-ccntilhons  (110  ,  Undec  mo- 
centill  ons i (111),  Duodecimo-centillions  (112),  Tertio-decimo-ccntillions  (113), 
QuiSi-Sdma2entUlion8  (114),  Vigesimo-centillions  (120),  M""^™- 
centillions  (121),  Trigesimo-centillions  (130),  Quadragesimo-ccntillionb  (140), 
Quinragesimo-centillions  (150),  Sexagesimo-centillions  (160),  Septuascsi- 
mo-centillions   (170),  Octogesimo-centillions  (180),   Nonagcsimo-centdhons 

tm ,D^^ 

Quimrentillions  (500),  Sexcentillions  (600),  Septingentil lions  (« 00),  Oct  - 
;SC (800),  Nongentillions  (900),  Millillions  (1000)  Centesimo-mill- 
mkms (1100),  Ducentcsimo-millillions  (1200),  Trecentesimo-m.llill.ons 
(1300) ,  Quadringentesimo-millillions(1400) ,Qu  ngentcsimo-iuil .11, ons  (1500) , 
Sexcentetimo-millillions  (1600),  Septingentesimo  -mi  hi  ions ("W^ggJ- 
o-PnteRimo-millillions  (1800),  Nongentesimo-millillions  (1900),  Bi-mi  ons 
ffl5£B\4,  Qutdri-millillions  (4000),  Qu J^^UIUIiodi 
HoOu  ;  S^exi  millillions  (6000)  Septi-millillions  (7000)  Octi-m HUhoiia  (8000  , 
Novi-millillions  (9000),  Deci-millillions  (10,000),  Undeci-millillions  (H,000), 
DnodSi-m  lillions  (12,000),  Tredeci-millillions  (13  000), .Qy^uonlwi-nnllp 
Ss  14  000),  Quindeci-millillions  (15,000),  Sexdeci-mill.lhons  (Ih.OW  ), 
K-d^tS  ions  (17,000),  Octi-deci-millillions  (UMJM).  ^^:m\  " 
ilfions  m  00(»  Vici-millillions  (20,000),  Semeli-vici-millill.ons  01.OM),  -BL 

SmillnS 

?24  So)  Quinqui-vici-millillions  (25,000),  Sexi-vici-mi  hlhons  »), 
Setti^ci-millllions  (27,000),  Octi-vici-millillions  (28,000),  Novi^ci-mill fft- 
SKSlSSTriSti^mitoM  (30,000),  Quadrasri-millillions  (40,000  ,Quin- 
SimffiLsT^OOO),  Sexagi-millillions  (60,000),  Septuagi-mil  llions 
?TOwSiBon    (80,000),  Nonagi-millillions  (90,000)    Centi-miU- 

Nongenti-millUlions  (900,000),  Milli-milli  ions  (1,^).  number8  to  be 
It  should  be  observed  that  words  ending  in  o  represent  numbers  to  ne 
added  and  those  endm? in  i  represent  multipliers.  When  two  words  end  in 
^sumVAhe^Smbe^  Indicted  is  to  be  »*^n  asthem«ltjpl-r.  In  each, 
the  last  word  indicates  the  number  to  be  increased  or  multiplied. 


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